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Electron-Electron Interactions in Two-Dimensional Spin-Orbit Coupled Systems

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Title:
Electron-Electron Interactions in Two-Dimensional Spin-Orbit Coupled Systems
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1 online resource (86 p.)
Language:
english
Creator:
Ashrafi, Ali
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Maslov, Dmitrii
Committee Members:
Tanner, David B
Ingersent, J Kevin
Hirschfeld, Peter J
Dranishnikov, Alexander Nikolae

Subjects

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

Notes

Abstract:
The work presented in this dissertation attempts to bring together two very active research venues in condensed matter physics. The rst one is the eld of strongly correlated electron systems and quantum phase transitions in itinerant electron systems. The second one is the eld of semiconductor spintronics and quantum computation, in particular, a subeld of the latter that studies coherence in semiconductor spin qubits. To explore interesting overlap regions between these two venues, we extend Landau's Fermi liquid (FL) theory to include a spin-orbit (SO) coupling. It is shown that although "charge-part" quantities, such as the charge susceptibility and eective mass, are determined solely by the quasi-particles, "spin-part" quantities, such as the spin susceptibility, have contributions from the damped states in between the two spin-split Fermi surfaces. Properties of such FL are discussed in detail. We also study the linear response of a SU(2) symmetric FL to a SO perturbation and predict new types of collective modes in FLs with SO coupling; chiral spin-waves. We nd the equations of motion for these modes and propose an experimental setup suitable to observe them.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Ali Ashrafi.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Maslov, Dmitrii.

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UFRGP
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Applicable rights reserved.
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lcc - LD1780 2013
System ID:
UFE0045900:00001

MISSING IMAGE

Material Information

Title:
Electron-Electron Interactions in Two-Dimensional Spin-Orbit Coupled Systems
Physical Description:
1 online resource (86 p.)
Language:
english
Creator:
Ashrafi, Ali
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:
Maslov, Dmitrii
Committee Members:
Tanner, David B
Ingersent, J Kevin
Hirschfeld, Peter J
Dranishnikov, Alexander Nikolae

Subjects

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

Notes

Abstract:
The work presented in this dissertation attempts to bring together two very active research venues in condensed matter physics. The rst one is the eld of strongly correlated electron systems and quantum phase transitions in itinerant electron systems. The second one is the eld of semiconductor spintronics and quantum computation, in particular, a subeld of the latter that studies coherence in semiconductor spin qubits. To explore interesting overlap regions between these two venues, we extend Landau's Fermi liquid (FL) theory to include a spin-orbit (SO) coupling. It is shown that although "charge-part" quantities, such as the charge susceptibility and eective mass, are determined solely by the quasi-particles, "spin-part" quantities, such as the spin susceptibility, have contributions from the damped states in between the two spin-split Fermi surfaces. Properties of such FL are discussed in detail. We also study the linear response of a SU(2) symmetric FL to a SO perturbation and predict new types of collective modes in FLs with SO coupling; chiral spin-waves. We nd the equations of motion for these modes and propose an experimental setup suitable to observe them.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Ali Ashrafi.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Maslov, Dmitrii.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045900:00001


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ELECTRON-ELECTRONINTERACTIONSINTWO-DIMENSIONALSPIN-ORBITCOUPLEDSYSTEMSByALIASHRAFIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013AliAshra 2

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ACKNOWLEDGMENTS Iwouldliketoexpressmysinceregratitudetomysupervisor,Prof.DmitriiMaslov,forhiscontinuedsupportandguidancethroughouttheprocessofthiswork.Notonlyhehasaveryvastknowledgeanddeepunderstandingofthesubject,heisalsoagreatmentor.Iamthankfulforallthestimulatingdiscussionswithotherfaculty,students,andvisitorsattheUniversityofFlorida,inparticularwithProf.AlexanderFinkelstein,Prof.PeterHirschfeld,Prof.JohnKlauder,Prof.PradeepKumar,Dr.HridisPal,Prof.EmmanuelRashba,Prof.SergeiShabanov,andDr.ChungweiWang.FinancialsupportfromthePhysicsdepartment,UniversityofFlorida,andtheNSFgrantDMR0908029,isalsoacknowledged.Finally,Iwouldliketousethisopportunitytothankmyparents,ParirokhandHasan,whoprovidedtheenvironmentformetogrowupasathinker. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFFIGURES ..................................... 6 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION:SPIN-ORBITINTERACTIONINCONDENSEDMATTERSYSTEMS ...................................... 8 2THEORYOFCHIRALFERMILIQUIDS ...................... 16 2.1Introduction ................................... 16 2.2LandauFunction ................................ 18 2.3EffectiveMass ................................. 21 2.4ThermodynamicProperties .......................... 27 2.4.1Compressibility ............................. 28 2.4.2Stabilityconditions ........................... 29 2.4.3SpinsusceptibilityofachiralFermiliquid ............... 30 2.5Zero-SoundCollectiveModesandPerturbationTheoryfortheLandauFunction ..................................... 35 2.5.1RelationbetweentheLandaufunctionandmicroscopicvertices .. 35 2.5.2Landaufunctionfromperturbationtheory .............. 37 2.5.3Example:massrenormalization ................... 43 2.5.4Perturbationtheoryforzz ...................... 48 2.5.4.1Weakelectron-electroninteraction ............. 48 2.5.4.2Weakspin-orbitcoupling .................. 49 2.6ChiralSpinWaves ............................... 50 2.6.1Dispersionofchiralspin-waves .................... 50 2.6.2Equationsofmotionforchiralspin-waves .............. 55 2.6.3Standingchiralwaves ......................... 61 2.6.4Parabolicconnementofchiralspin-waves ............. 63 2.6.5Conditionsforobservation ....................... 64 2.6.6Excitationofchiralspin-waves ..................... 65 3CONCLUSIONS ................................... 68 APPENDIX AMASSRENORMALIZATIONVIATHESELF-ENERGY .............. 71 BOUT-OF-PLANESPINSUSCEPTIBILITYOFNON-INTERACTINGRASHBAELECTRONS:ATHERMODYNAMICCALCULATION .............. 76 4

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CKOHNANOMALYINAFREEELECTRONGASWITHRASHBASPIN-ORBITCOUPLING ...................................... 78 DNON-ANALYTICCONTRIBUTIONSTOTHELANDAUFUNCTION ....... 80 REFERENCES ....................................... 82 BIOGRAPHICALSKETCH ................................ 86 5

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LISTOFFIGURES Figure page 2-1TheDysonequationforthescatteringvertex. ................... 36 2-2Allsecond-orderdiagramsforthe)]TJ /F6 7.97 Tf 6.78 4.34 Td[(vertexrelatedtotheLandaufunction. .. 38 2-3Selfenergydiagramstorstorderintheinteraction. ............... 48 2-4SpectrumofchiralspinwavesforFa,0=)]TJ /F5 11.955 Tf 9.3 0 Td[(0.5. .................. 54 2-5Sketchofthesuggestedexperimentalsetup. ................... 63 C-1Polarizationbubble .................................. 78 6

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyELECTRON-ELECTRONINTERACTIONSINTWO-DIMENSIONALSPIN-ORBITCOUPLEDSYSTEMSByAliAshraAugust2013Chair:DmitriiMaslovMajor:PhysicsTheworkpresentedinthisdissertationattemptstobringtogethertwoveryactiveresearchvenuesincondensedmatterphysics.Therstoneistheeldofstronglycorrelatedelectronsystemsandquantumphasetransitionsinitinerantelectronsystems.Thesecondoneistheeldofsemiconductorspintronicsandquantumcomputation,inparticular,asubeldofthelatterthatstudiescoherenceinsemiconductorspinqubits.Toexploreinterestingoverlapregionsbetweenthesetwovenues,weextendtheLandau'sFermiliquid(FL)theorytoincludeaspin-orbit(SO)coupling.Itisshownthatalthoughcharge-partquantities,suchasthechargesusceptibilityandeffectivemass,aredeterminedsolelybythequasi-particles,spin-partquantities,suchasthespinsusceptibility,havecontributionsfromthedampedstatesinbetweenthetwospin-splitFermisurfaces.PropertiesofsuchFLsarediscussedindetail.WealsostudythelinearresponseofanSU(2)symmetricFLtoanSOperturbationandpredictnewtypesofcollectivemodesinFLswithSOcoupling:chiralspin-waves.Wendtheequationsofmotionforthesemodesandproposeanexperimentalsetupsuitabletoobservethem. 7

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CHAPTER1INTRODUCTION:SPIN-ORBITINTERACTIONINCONDENSEDMATTERSYSTEMSManymaterialsofcurrentinterestarecharacterizedbynon-trivialcorrelationsbetweenthespinandorbitaldegreesoffreedom.Tonamejustafew,thesearetwo-dimensional(2D)electronandholegasesinsemiconductorheterostructureswithbrokeninversionsymmetry[ 1 ],surface/edgestatesofthree-dimensional(3D)/2Dtopologicalinsulators[ 2 4 ],conducting(andsuperconducting)statesatoxideinterfaces[ 5 ],etc.Spin-orbit(SO)couplinginherenttoallthesesystemslockselectronspinsandmomentaintopatternscharacterizedbychirality,andwewillrefertomaterialsofthistypeastoelectron-chiralmaterials.1Electron-chiralmaterialsareendowedwithuniquepropertiesthatareinterestingfromboththefundamentalandappliedpoints-of-view.Onthefundamentalside,electron-chiralmaterialsprovideanideallaboratorytostudysuchimportantconsequencesofmagneto-electriccouplingasthespin-Halleffect[ 7 9 ]andspintextures(helices)[ 10 12 ].Ontheappliedside,thesematerialsarestudiedaspotentialplatformsforspintronicdevicesthatallowelectricmanipulationofmagneticproperties.Inaddition,thetopologicalsubclassofelectron-chiralmaterialsisbeingactivelyinvestigatedasaplatformfornon-Abelianquantumcomputation[ 13 ].ThepossibilityoffullcontroloverelectronspinentirelyviaelectricmeansisthemainreasonwhySOcoupledsystemsareingeneralamongthemostinterestingsystemsunderresearch.Therefore,itisinstructivetostudydifferenttypesofSOinteractions.Inwhatfollows,abriefintroductiontoSOinteractionsisgiven,withafocusonlowdimensions. 1PseudospinsofDiracfermionsingraphenearealsocorrelatedtotheirmomenta,andthusgraphenecanbeviewedasapseudochiralsystem[ 6 ]. 8

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SOcoupling,havingitsorigininrelativisticquantummechanics,wasrstobservedinatomicphysics.Theeffectariseswhenelectronsoftheatom,intheirrestframe,experienceamagneticeldwhileorbitingthechargednucleus.ThiseffectiscapturedmainlyinthePauliSOtermintheexpansionoftheDiracequation,andalsogeneratesSOcouplingincondensed-matter(CM)systems.OnewouldclassifytheSOCinCMsystemsaseithersymmetry-independentorsymmetry-dependent.Whiletheformerislargelythesameasinatomicphysicsandexistinalltypesofcrystals,thelatterisonlypresentinsystemswithbrokeninversionsymmetry.Dependingonthetypeofbrokeninversionsymmetry,whetherinthebulkorthesurface,theeffectgoesundergenericnamesofDresselhausorRashba,respectively.UnliketheDresselhausSOinteraction,thestructureoftheRashbatermdoesnotdependonthesymmetryofthehostcrystalandhence,itissimplertoconstructtheRashbaHamiltonianongeneralgrounds.Inwhatfollows,rstaderivationoftheRashbaSOHamiltonianisgiven,andothertypesofSOinteractionarediscussedbrieylater.InsteadofadetailedderivationoftheSOHamiltonian,startingwiththeoriginalPauliSOterm,onecanexploretheclassofHamiltoniansthatallowforcouplingbetweenthespinandmomentumgivencertainsymmetries.Considerasingle-particleHamiltonianthatdependsonlyonthemomentumandthespinoftheparticleandpreservestimereversal,SO(2)rotationalsymmetryandreectioninverticalplanes.2Giventhesesymmetries,andthefactthathigherthanlinearpowersofthespinoperatorareabsent,themostgeneralSOHamiltonianisoftheform[ 14 ] i(p2n+1)]TJ /F3 11.955 Tf 20.91 2.96 Td[(+)]TJ /F7 11.955 Tf 11.96 0 Td[(p2n+1+)]TJ /F5 11.955 Tf 7.08 1.8 Td[(),(1) 2AdirectproductofSO(2)andreectiongroupsisisomorphictothegroupC1v. 9

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wherea=axiay,isaSOcouplingconstant,andrepresentsprojectionsofjz=(n+1 2).Toprovetherotationalinvariance,weexaminethetransformationofsuchatermunderanSO(2)elementR(rotationof2Drealvectorsbyanangle), R~p=0B@cos)]TJ /F5 11.955 Tf 11.29 0 Td[(sinsincos1CA0B@pxpy1CA. (1) Therefore, Rp=eip,(1)andhence, Rp2n+1=ei(2n+1)p2n+1.(1)Thespinsmustbetransformedaccordingly R=ei(2n+1).(1)Thisisidenticaltotransformingtheexpectationvalueofthespins.3Thereforetherearetworotationallyinvarianttermsavailable:p2n+1)]TJ /F3 11.955 Tf 20.91 2.61 Td[(+,andp2n+1+)]TJ /F1 11.955 Tf 7.08 1.8 Td[(.Therelativeprefactorbetweenthetwotermsisdeterminedbyenforcingthein-planereectionsymmetry;forinstance,reectioninthey)]TJ /F7 11.955 Tf 12.07 0 Td[(zplaneresultsink!)]TJ /F7 11.955 Tf 24.86 0 Td[(kand!,andhenceonearrivesatthelinearcombinationinEq. 3ThisistobedistinguishedfromrotationofthespinsinthespinspaceRspinji=ei(n+1 2)ji. 10

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InEq. 1 ,thecaseofn=0describestheRashbaSOinteraction[ 15 ]forelectronswithspin1=2,whichcanbewrittenas HR=(p)ez,(1)whereezisthenormalvectoroutoftheplaneofthesystem,andisthelinearRashbaSOcouplingconstant.TondthestrengthoftheSOcoupling,onehastostartbyaddingthePauliSOtermtothefullnon-relativisticHamiltonianofasingleelectroninsideasolid,includingexternalconningpotentialsanduseapproximatemethodstocapturetherelevantSOterms.Forinstanceinthecaseofacubiccrystal,aminimaleffectivetheorycontains8-bandsinthekptheory(theso-called88Kanemodel)andEnvelopeFunctionApproximationsuitableforslowlyvarying(comparedtothelatticespacing)externalpotentials.TheleadingorderresultfortheRashbatermforelectronsinacubiccrystal,producedbythisscheme,is: ~rV jrVj(p)=P2 31 E2g)]TJ /F5 11.955 Tf 37.68 8.09 Td[(1 (Eg+0)2rV(p),(1)whererV=eEz^zistheexpectationvalueoftheconningelectriceldEzinthevalenceband(thisisasubtlebutfundamentalpoint;seeSection6.3.2ofRef.[ 16 ])timesthechargeoftheelectrone,Egand0arethebandgapandSOsplittingofthevalenceband,respectively,andPisamatrixelementofthemomentumintheconningdirection.ItisworthmentioningthattheRashbaspinsplittingforholesistotallydifferentfromthatforelectrons.Forinstance,inthesamemodel,asimilarexpressionfortheRashbacouplingforholesincludesadifferentcombinationofbandstructureparametersthantheoneinsidethebracketsinEq. 1 ;forathoroughdiscussionsee[ 16 ].Thenaturalunitfor~ismeVA;whenmeasuredintheseunits~isoforderunityinmostsemiconductors.Forinstance,~=3.04meVAinGaAs,while~=245.7meVAinInSb[ 16 ].Hence,fornumberdensityn=1012cm)]TJ /F6 7.97 Tf 6.58 0 Td[(2,theSOenergyscale 11

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pF=0.07meVinGaAsandpF=6.14meVinInSb,wherepF=p 2nistheFermimomentum.ThemostcommonwaytoexperimentallymeasureSOcouplingisbythestudyofShubnkovdeHaasoscillationsatsmallmagneticeldsandlowtemperatures,inwhichbeatingintheseoscillationssignalsthedifferenceinspinsub-bandpopulations.AnothertypeofstructuralSOinteractionistheDresselhausSO,whichoriginatesfromalackofinversionsymmetryinthebulkofmaterial,andcausesspin-splittingofthebulkconductionband.The2DformoftheDresselhausinteractiondependsonthedirectionoftheconnementplane.Forinstance,inthecaseofaconnementinthe(001)direction,theso-calledlinearDresselhausSOisgivenby[ 17 ] HD=(xpx)]TJ /F3 11.955 Tf 11.95 0 Td[(ypy),(1)whereistheDresselhauscouplingconstant.Asisclearfromtheexpression,aslongastherestoftheHamiltonianisisotropic,anysystemwithonlyoneofthetwo(RashbaorDresselhaus)couplingsisequivalenttoasystemhavingonlytheother,sincetheyarerelatedbyaunitarytransformation.However,asystemwithhavingbothcouplingscannotbetransformedintoasystemhavingonlyone.TherearealsosystemsinwhichtheorbitalangularmomentumenterstheSOinteractionaswell.Awell-knownexampleofsuchsystemisholesinGaAs,whichareformedbythestateswithorbitalmomentuml=1.Inthebulk,thebandsofheavyholeswithjz=3=2andoflightholeswithjz=1=2aredegenerateatk=0.Inaquantumwell,degeneracyisliftedandthelight-holebandissplitofftheheavy-holeone.IftheFermienergylieswithintheheavy-holeband,wehaveasystemwithpseudospinjz=3=2.AparticularformoftheSOcouplinginsuchasystemisthecaseofn=2inEq. 1 HR3=i3(p3)]TJ /F3 11.955 Tf 7.08 2.96 Td[(+)]TJ /F7 11.955 Tf 11.95 0 Td[(p3+)]TJ /F5 11.955 Tf 7.08 1.79 Td[(),(1) 12

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where3isaSOcouplingconstantthatcanbefoundinasimilarfashiontothelinearcoupling[ 16 ].Thisistheso-calledcubicRashbainteraction.Inclusionofanin-planemagneticeldinsuchasystemsisnottrivialbecauseoftheparticularrotationpropertiesoftheeffectivespin3=2.Magneticeldisapseudovector,i.e.,rotateslikeavectorbutreectswithanadditionalsigncomparedtoavector.Hence,tolinearorderinB=BxBy,onecanconstructthefollowingHamiltonianconsistentwiththesymmetrygroupC1v: HB3=03(p2)]TJ /F7 11.955 Tf 7.09 2.96 Td[(B)]TJ /F3 11.955 Tf 7.08 1.79 Td[(++p2+B+)]TJ /F5 11.955 Tf 7.08 1.79 Td[(),(1)where03isacouplingconstantthatagaincanbedeterminedfromthekptheory[ 14 ].Inadditiontoheavyholesinthe(001)quantumwellinAlGaAs/GaAs,thisHamiltonianisrelevantforthe(001)surfacestateinSrTiO3[ 18 ]and,possibly,forthe2DsuperconductingstateattheLaAlO3=SrTiO3interface(LAO/STO)[ 19 ].4Whilethesingle-particlepropertiesofelectron-chiralsystemsarewellunderstoodbynow,weareonlystartingtoappreciatetheimportanceoftheinterplaybetweenSO/chiralityandtheelectron-electron(ee)interaction,whichisnecessarilypresentinallthesesystems.Theeeinteractionisthedrivingforceofphasetransitions(betheynite-orzero-temperature)inelectronsystems.Forexample,afamiliarferromagnetic(Stoner)instabilityintheHubbardmodeloccursbecausetheincreaseofthekineticenergyduetospontaneousspinpolarizationisoffsetbythedecreaseoftheinteractionenergydueto(partialorcomplete)eliminationofminority-spinelectrons.Inourcurrentpictureofferromagneticandothermagneticinstabilitiesinelectronsystems,theSOinteractionplaysanimportantbutsecondaryrole:forexample,ithelpstopinthedirectionofthemagnetizationinaferromagnetbutdoesnotalterthepropertiesof 4AlthoughtheanalysisofquantummagnetoresistanceinLAO/STOwascarriedoutassumingalinearRashbacoupling[ 20 ],ademonstratedcubicRashbacouplingintheparentmaterial[ 18 ]suggestthatacubiccouplingisalsorelevantforLAO/STO. 13

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theferromagneticphaseitself.ThispictureworksiftheSOenergyisthesmallestenergyscaleintheproblem.However,thecurrenttendency)]TJ /F1 11.955 Tf 12.62 0 Td[(driveninpartbytheneedtodevelopsemiconductor-basedspintronicdevices)]TJ /F1 11.955 Tf 9.3 0 Td[(istoenhancethestrengthoftheSOinteractionrelativetootherimportantscales,namely,theFermienergy,andCoulombrepulsion[ 21 ].5Ifthesethreeenergyscalesarecomparabletoeachother,weventureintoanalmostunexploredregimewhereeeinteractioncandrivethesystemintoamagnetically-orderedstate,yetstrongSOcouplingmakesthisstatequalitativelydifferentfromfamiliarferro-andanti-ferromagnets.Forexample,astrongenoughCoulombrepulsioncanforceallelectronstooccupyonlyoneRashbasubband[ 14 29 ]ortoformaspin-nematicwithaspin-splitFermisurfacebutzeronetmagnetization[ 30 32 ].GiventhepresenceofverystrongSOcoupling(comparedtotheFermilevel)inmaterialssuchasinBiTel6,p-GaAswithlowcarrierdensity,InSb,etc.,andstrongeecorrelationsinsomeofthesesystems,e.g.,heavy-holep-GaAs7,atheorythatisnon-perturbativeeitherinSOorineeinteractions,canprovetobeveryuseful.InChapter2ofthisdissertation,weconstructageneraltheoryofchiralFermiliquids(FLs).Itishardtooverestimatetheimportanceofsuchatheory.Thewell-establishedtheoryofnon-chiral(orSU(2)-invariant)FLs[ 34 36 ]allowsonetoclassifypossible 5Insemiconductorheterostructures,thiscanbeachievedbyreducingthecarriernumberdensitytothepointwhenonlythelowestofthespin-splitRashbasubbandsisoccupied.ThispossibilityhasbeendemonstratedintheprocessofsearchingforMajoranafermionsinp-GaAs[ 22 ],InSb[ 23 ],andInAs[ 24 ]quantumwires.Inothersystems,suchasagraphene/1mlAu/Niheterostructure[ 25 ]andsurfacestatesofbismuthtellurohalides(studiedatUF[ 26 ])[ 27 28 ],theSOcouplingisincreasedduetothepresenceofheavy(AuandBi)atoms.6FermionsatthesurfaceBiTeldifferfromelectronsin,e.g.,n-GaAsheterostructures,bytheabsenceofthequadraticterminthespectrum.Consequently,onlyoneRashbasubbandinthissystemisoccupied.7Parameterrsreaches10-40inp-GaAsheterostructures[ 33 ]. 14

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PomeranchukinstabilitiesthatbreakrotationalbutnottranslationalsymmetryoftheFermisurfaceandderivethestabilityconditionsintermsoftheLandauparameters.Italsoprovidesfullclassicationandnon-perturbativetreatmentofcollectiveexcitationsinthechargeandspinsectors(whicharedecoupledinthenon-chiralcase).Withtheexceptionofone-dimensional(1D)systemsandsurfacestatesof3DtopologicalinsulatorsattheDiracpoint,allotherexamplesofelectron-chiralelectronmaterialsarebelievedtobehaveasFLswithwell-denedquasi-particlesneartheFermisurface(ormultipleFermisurfacesincaseofspin-splitstates).ThisconclusionissupportedbyanumberofstudiesinwhichvariousFLquantities)]TJ /F1 11.955 Tf 9.3 0 Td[(effectivemass,quasi-particlelifetime,spinsusceptibility,spin-Hallconductivity,spin-Coulombdragcorrectiontothechargeconductivity,plasmonspectrum,etc.)]TJ /F1 11.955 Tf 9.3 0 Td[(werecalculatedwithinperturbationtheorywithrespecttotheeeinteraction[ 14 29 37 50 ].Inaddition,severalpreviousattempts[ 51 53 ]toconstructFLtheorieswithSOcouplingarediscussedfurtherinthenextChapter. 15

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CHAPTER2THEORYOFCHIRALFERMILIQUIDS 2.1IntroductionBeingthereasonformanyspin-dependentphenomenaparticularlyinlowdimensionalsystems,theeffectsofSpin-Orbit(SO)interactionremainsasubjectofintenseresearch.AninterestingareaistheroleofeeinteractioninsystemswithSOcoupling.Aclassicproblemthatseemstohavebeenneglectedpreviously,istheconstructionofaFermiliquid(FL)theoryforsystemswithSOcoupling.Landau'sFLtheorydescribesthesystemasanensembleofalmostfreeexcitations,i.e.,quasiparticles(QPs),neartheFermisurface(FS)butwithasetofrenormalizedparameters,suchastheeffectivemassandLandegfactor.High-energyphysics,i.e.,physicsofstatesawayfromtheFermienergywhichisformidabletoaccountforinperturbationtheory,isencapsulatedinanumberofphenomenologicalparameters.GiventhesuccessofFLtheoryindescribingthepropertiesinmanyfermionicsystems(He3,simplemetalsandtheiralloys,degeneratesemiconductors,coldfermionicatoms),itisdesirabletoextendthetheorytoincludetheSOinteraction.WeshallonlyconsideralinearRashbacouplingintwodimensional(2D)FLs,asthesimplestSOcouplingpossible.Somequasiparticleparameters,suchasthelifetime,Z-factor,andeffectivemass,havebeenpreviouslycalculatedincludingthelong-rangeCoulombinteractionwithintherandomphaseapproximation(RPA)[ 48 ].SuchatheorycanshedlightonclassicationofpossiblePoemranchuckinstabilities(correspondingtospatiallyuniformphases)inthepresenceofSOinteractions.PerturbativestudiesmakeupasignicantbodyofexistingliteratureontheeeinteractionsinSOcoupledsystemssuchaspossiblerstorderphasetransitionsinsuchsystems[ 29 54 ],studiesofthefermionicself-energy[ 55 ],spatiallynon-uniformcontinuousphasetransitions[ 56 ],chargesusceptibilityandSOrenormalizationofFriedeloscillationsinthescreenedpotential[ 38 ],andspinsusceptibilities[ 44 45 ],tonameafew. 16

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ExplicitinclusionofSOintheframeworkofFLtheoryhaspreviouslybeenpresented,toourknowledge,onlyinRefs.[ 52 ]and[ 51 ].TheformerpreservesthespatialinversionsymmetryandconsidersgenerictensorinteractioninthreedimensionalFLs,andwearguethatthisworkmissesthesubtletiesarisingduetothebrokenSU(2)symmetryinducedbynon-spin-conservingforcessuchasSO.Reference[ 51 ]includestheeffectsoftheRashbaorDresselhausSOonlyasaperturbingforcetotheSU(2)-symmetric(SU2S)FLtostudyspinresonanceandspinHallconductivity.ThisworkaimstoaddressthequestionofwhetherthereisaFLtheoryforanelectronsystemwithaSOinteraction.Wearguethattheanswerdependsonwhetherthepropertyunderstudyisspin-independent,e.g.,chargesusceptibilitiesorspin-dependent,e.g.,spinsusceptibilities.Forspin-independentquantitieswithlocalpropertiesinSU(2)spinspace,theoriginalLandau'sFLtheorycanbeextendedtoincludetheeffectsofSOwithratherstraightforwardmodicationssimilartoanytwo-bandFLtheory.Forspin-dependentphenomenaontheotherhand,anyextensionofaconventionalFLtheorywouldfailasthesepropertiesoftheliquidarecontrolledbythedampedmany-bodystatesinbetweenthetwoFSs.Asimilarconclusionwasreachedforapartiallyspin-polarizedFL[ 57 ].However,toaccountforrstordereffectsinSO,onecanneglectthenitedistancebetweenthetwoFSsandkeepSOmerelyintheenergysplittinginducedbySO.ThisChapterisorganizedasfollows.Inthenextsection,weconstructtheLandauinteractionfunction(LF)asthemaintoolforphenomenologicalstudyoftheFL.EquippedwiththeLF,weproceedtocalculatetheeffectivemassesoftheFL,inSec. 2.3 .WediscussanOverhauser-typesplittingoftheeffectivemassessimilartoeffectivemasssplittinginapartiallyspin-polarizedmetal.ThecompressibilityoftheliquidandthePomeranchukstabilityconditionarederivedinSec. 2.4 .Zerosoundcollectivemodesareshowntobedeterminedbyasetofcoupledequationsforthescatteringamplitude.InSec. 2.5 weusetheseequationstorelatethephenomenologicaland 17

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microscopictheories.AnexplicitformoftheLFisevaluatedmicroscopicallyinamodelwithshortrangeinteractionsinSec. 2.5.2 ,conrmingtheformproposedinSec. 2.2 .Toillustratethesubtletyinthespinsector,theout-of-planespinsusceptibilityisevaluatedasanexampleinSec. 2.4.3 andtheimportanceofthecontributionofdampedstatesinbetweenthetwoFSsisshownexplicitly.Itisarguedthatthisfeatureispresentforallspin-dependentpropertiesofanyFLwithbrokenSU(2)symmetry,andhencetheyshouldbeconsideredasFLswithoutaconventionalFLtheory.InSec. 2.6 ,weconsideranewtypeofcollectivemodesinchiralFLs,namely,chiralspinwavesandproposeanexperimentinwhichtheycanbeobserved.Thesecollectivemodesarefoundinthespinsectorofthetheory,usingSOasaperturbationtotheSU2SFL. 2.2LandauFunctionWeconsidera2DsystemofelectronsinthepresenceoftheRashbaSOinteraction,describedbytheHamiltonian ^H=^Hf+^Hint=p2 2m1+(^p)ez+^Hint,(2)wheremistheeffectiveelectronmass,^arethePaulimatrices,ezistheunitvectoralongthenormaltothe2DEGplane,and^Hintentailsanon-relativistic,density-densityeeinteraction.(HereandintherestoftheChapter,ischosentobepositive,and~andkBaresettounity.Also,theindexfwilldenotethepropertiesofafreesystem.)ThespacegroupofHamiltonian 3 isC1v,andtheRashbaSOtermistheonlycombinationofthespinandmomentumthatisinvariantunderthesymmetryoperationsofthisgroupseeChapter 1 .Eigenvectorsandeigenenergiesof^Hfaregivenby js,pi=1 p 20B@1)]TJ /F7 11.955 Tf 9.3 0 Td[(iseip1CA(2)and p,fs=p2=2m+sp,(2) 18

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wherepisthepolarangleofthemomentump,ands=1denotesthechirality,i.e.thewindingdirectionofspinsaroundtheFS.IntherestofthisChapter,weassumethatthechemicalpotential,,intersectsbothRashbasubbands.The(bare)FermimomentaandFermivelocitiesoftheindividualsubbandsaregivenby pf=p (m)2+2mm=m(v0) (2) vfF=v0p 2+2=m. (2) InaSU2Stheory,theonlyrelevantobjectsinspinspacearespinsofthetwoQPsandtheinvarianttensor1(22Kronecker),outofwhichthetensorialstructureoftheLFcanbebuiltrespectingtheSU(2)symmetry, ^f(p,p0)=fs(p,p0)110+fa(p,p0)^^0,(2)writteninarbitraryspinbasis.Scalarcombinationsfs(fa)arethespin-symmetric(anti-symmetric)partsoftheLF,pandp0representmomentaofthetwoquasiparticles.Clearly,spinandchargepartsarecompletelyseparated.However,thispictureisincorrectuponinclusionofSO.ThecentralobjectoftheFLtheoryistheLandaufunctiondescribingtheinteractionbetweenquasiparticleswithmomentapandp0,andspins^and^0.SinceSOcouplingreducesSU(2)spinsymmetrydowntoU(1),theLandaufunctionwillcontainmoreinvariantscomparedtotheSU2Scase[Eq. 2 .ThetaskofndingallinvariantsissimpliedbynotingthattheRashbacouplingisequivalenttotheeffectofanon-AbelianmagneticeldBR(p)=(2=gB)(py,)]TJ /F7 11.955 Tf 9.3 0 Td[(px,0),wheregistheelectron'sg-factorandBistheBohrmagneton.ThemostgeneralformoftheLandaufunctionmustincludescalarproductsformedoutofthesixobjects1,^andBR(p)foreachofthetwoquasiparticlesthatareinvariantunderC1v,timereversal,andpermutationsofquasiparticles.TheLandaufunctionofaFLintherealmagneticeldBcontainsextraZeemanterms,^0Band^B,aswellastheirproducts.Inaddition 19

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tothesecouplings,theRashbaeldcanbecoupledtospinsinmorewaysbecause,incontrasttothereal-eldcase,thecrossproductBR(p)BR(p0)/^zjpp0jisnon-zeroandactsonspinsasanothereffectivemagneticeldalongthezaxis.Exploringallpossibleindependentinvariants,wearriveatthefollowingLandaufunction: ^f=fs110+fak(^x^0x+^y^0y)+fa?^z^0z+1 2gph1^0(p0)]TJ /F14 11.955 Tf 11.95 0 Td[(p)ez)]TJ /F5 11.955 Tf 11.96 0 Td[(10^(p0)]TJ /F14 11.955 Tf 11.95 0 Td[(p)ez+1 2gpp1^0(p0+p)ez+10^(p0+p)ez+h(1)(^^pez)(^0^pez)+h(2)(^^p0ez)(^0^pez)+1 2h(^^pez)(^0^pez)+(^^p0ez)(^0^p0ez), (2) wherethescalarfunctionsfsthroughhsharethesameargument(p,p0),whichweomitforbrevity.InSec. 2.5 ,weshowthatallthetermsinEq. 2 areproducedbytheperturbationtheoryfortheinteractionvertex.Thesuperscriptsphandppinthegtermsindicatethat,intheperturbationtheory,thecorrespondingtermscomefromtheinteractionintheparticle-holeandparticle-particle(Cooper)channels,respectively.Asistobeexpected,SOcouplingbreaksspin-rotationalinvarianceoftheexchange,^^0,termofanSU2SFL.Anisotropyintheexchangepartof^fcomesfromthecombination[^(BRB0R)]^0(BRB0R)/^z^0z,whichaffectsonlythe^z^0zpartoftheexchangeinteraction.[Here,BRBR(p)andB0RBR(p0).]Consequently,theanisotropicexchangeinteraction[rstlineofEq. 2 containsdifferentcouplings(fajjandfa?)forinteractionbetweenin-planeandout-of-planespins.Recentperturbativecalculations[ 14 44 45 ]showthatanisotropyisoftheIsingtype,i.e,thatfa?>fajj.Inadditiontobreakingtherotationalsymmetryoftheexchangeinteraction,theSO 20

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couplinggeneratestheeffectiveZeemanterms[withcouplingconstantsgph,gpp,h(1),andh(2)],whichdependexplicitlyonpandp0.Itisworthpointingoutthatthehtermscanbewrittenintwoequivalentforms.Forexample,theh(1)termcanbewrittendowneitheras(^pez)(^0p0ez)oras(^p)(^0p0),whichresultsinequivalentLFsuponre-deningthescalarfunctionfak.SpinsinEq. 2 arenotyetrepresentedinanyparticularbasis.However,inordertoprojectelectrons'momentaontheFSs,weneedtospecifythebasis.Sincetheeeinteractioncommuteswithspins,thespinstructureofquasiparticles'statesisstillgovernedbytheRashbaterm,aslongastheinteractionisbelowthecriticalvalueforaPomeranchukinstability.Thisargumentisnothingmorethantheusualassumptionthatsymmetriesofthesystemdonotchangeiftheinteractionisswitchedonadiabatically.MicroscopiccalculationsinSec. 2.5 indeedshowthatthespinstructureofquasiparticlesisthesameasoffreeelectrons.Therefore,wetakethechiralbasisofEq. 2 astheeigenbasisforquasiparticlesofaRashbaFL. 2.3EffectiveMassLandau'sderivationoftheFLeffectivemassisrestrictedtoGalileaninvariantsystems,andthuscannotbeappliedtoourcasebecausetheSOtermbreaksGalileaninvariance.AgeneralizationoftheLandau'sderivationforarelativisticFL[ 58 ]isalsonotapplicableherebecauseourHamiltonianEq. 3 isnotfullyLorentzinvarianteither.Therefore,weneedtodeviseanargumentwhichinvolvesneitherGalileannorLorentzboosts.Tothisend,wenoticethat,since^Hintdependsonlyonthepositionsofelectronsbutnotontheirvelocities,thepositionoperatorcommuteswith^Hint.Therefore,thevelocityoperatoristhesameasintheabsenceoftheeeinteraction: ^vj=)]TJ /F7 11.955 Tf 9.3 0 Td[(i[^xj,^H]=)]TJ /F7 11.955 Tf 9.3 0 Td[(i[^xj,^Hf]=pj m1+(ez^).(2) 21

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SummingEq. 2 overallparticlesgives TrXjvj=1 mTrXj[pj1+m(ez^)].(2)Forthenon-relativisticcase(vjc,8j)consideredhere,theleft-handsideofEq. 2 isthetotaluxofparticles.(Relativisticcorrectionstothisresultareoforderv2=c2,whichisthesameorderastherelativisticcorrectiontothespectrum,assumedtobemuchsmallerthantheSOcorrection.)Bydenition,thetotaluxofparticlesisequaltothatofquasiparticles.Thesameistrueforthetotalmomentumandtotalspin.Hencethesumsoverparticlescanbeconvertedintosumsoverquasiparticles.Usinga22occupationnumberofquasiparticles^np,wehave TrZ[p1+m(ez^)]^np(dp)=mTrZ@p^"p^np(dp),(2)where(dp)d2p=(2)2,^"pistheenergyfunctionalforquasiparticles,and@p^"pistheirvelocity.SinceboththeHamiltonianand^nparediagonalinthechiralbasis,Eq. 2 inthechiralbasisreads XsZ[p+m(ez^ss)]nps(dp)=mXsZ@p^"psnps(dp), (2) wherenpsnpss.Nowweapplyarbitraryindependentvariationsonthediagonalelementsofthedensitymatrix,keepingoff-diagonalelementstobezero:nps=n0ps+nps,wheren0ps=(ps)]TJ /F7 11.955 Tf 12.52 0 Td[(p)atT=0andpsistheFermimomentumoffermionswithchiralitys.(Here,superscript0referstotheunperturbedFLwithnovariationsintheoccupationnumber.)Avariationofthediagonalelementofthequasiparticleenergy"ps"pssisrelatedtonpsvia "ps=Xs0Zfs,s0(p,p0)np0s0(dp0),(2)wherefs,s0(p,p0)fs,s0;s,s0(p,p0)andfs1,s2;s3,s4(p,p0)istheLandaufunctionfromEq. 2 inthechiralbasis(alltheunprimedPaulimatricescomewithwiths1s3andalltheprimed 22

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oneswiths2s4).VaryingbothsidesofEq. 2 ,weobtain XsZ[p+m(ez^ss)]nps(dp)=mXsZ@p"0psnps(dp)+mXsZ@p"psn0ps(dp). (2) TheLFisgivenby fs1,s2;s3,s4=fs1s1s310s2s4+fak(^x^0x+^y^0y)+fa?^z^0z+1 2gph1^0(p)]TJ /F14 11.955 Tf 11.95 0 Td[(p0)ez)]TJ /F5 11.955 Tf 11.95 0 Td[(10^(p0)]TJ /F14 11.955 Tf 11.96 0 Td[(p)ez+1 2gpp1^0(p+p0)ez+10^(p+p0)ez+h1(^epez)(^0ep0ez)+h2(^ep0ez)(^0epez)+1 2h(^epez)(^0epez)+(^ep0ez)(^0ep0ez), (2) whereep=p=p.Integrationbypartsinthersttermoftheright-handsideofEq. 2 gives,uponrelabelings !s0andp !p0andusingthesymmetryfs,s0(p,p0)=fs0,s(p0,p), XsZ[p+m(ez^ss)]nps(dp) (2) =mXs(@p"0ps)nps)]TJ /F7 11.955 Tf 11.96 0 Td[(mXs,s0Zfs,s0(p,p0)(@p0n0p0s0)nps(dp)(dp0).Sincevariationsarearbitrary,theintegrandsthemselvesmustbeequal.Using@p0n0p0s0=)]TJ /F3 11.955 Tf 9.3 0 Td[((ps0)]TJ /F7 11.955 Tf 12.03 0 Td[(p0)ep0and@p"0ps=ps=msasadenitionofthequasiparticleeffectivemass(withpspsp=ppsep),projectingEq. 2 ontoepandsettingp=ps,weobtain ps1+m ps(ez^ss)ep=psm ms+mXs0Zfs,s0(ps,p0)(ps0)]TJ /F7 11.955 Tf 11.95 0 Td[(p0)epep0(dp0), (2) 23

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Tosimplifythenalformofms,wedene Fs,s0(p,p0)=sfs,s0(p,p0)(2)withs=ms=2beingthedensityofstatesofthesubbands.Allthescalarfunctionsinthe^fmatrixcanbere-denedinthesamefashion,e.g. Fs(p,p0)=sfs(p,p0) (2) andsimilarlyfortheFak,Fa?,Gph,Gpp,H(1),H(2),andHfunctions.Withthesedenitions,Eq. 2 canbewrittenas ms m1+m ps(ez^ss)ep=1+Xs0ps0 psF,1ss0,(2)where F,`s,s0ZFs,s0(ps,ps0)cos(`pp0)dpp0 2(2)andp,p0istheanglebetweenpandp0.UsingexplicitformsofthePaulimatricesinthechiralbasis 2 ^x=0B@sinpicosp)]TJ /F7 11.955 Tf 9.3 0 Td[(icosp)]TJ /F5 11.955 Tf 11.29 0 Td[(sinp1CA,^y=0B@)]TJ /F5 11.955 Tf 11.29 0 Td[(cospisinp)]TJ /F7 11.955 Tf 9.3 0 Td[(isinpcosp1CA,^z=0B@01101CA,(2)wend(ez^ss0)ep=sss0.ProjectingEq. 2 ontothechiralbasis,weobtain Fs,s0(ps,ps0)=Fs(ps,ps0)+Fak(ps,ps0)ss0cospp0+Gph(ps,ps0)[sps+ss0)]TJ /F5 11.955 Tf 11.96 0 Td[((sps0+ss)cospp0]=2+Gpp(ps,ps0)[sps+ss0+(sps0+ss)cospp0]=2+H(1)(ps,ps0)ss0cos(2pp0)+H(2)(ps,ps0)ss0cos2pp0+H(ps,ps0)ss0cospp0. (2) Weseethat,incontrasttothecaseofaSU2SFL,whentheeffectivemasscontainsonlythe`=1harmonicofFs,the`=1harmonicofFss0inEq. 2 containsalso 24

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higherharmonicsofpartialcomponentsofFs,s0;namely,itcontainsthe`=2harmonicsofFak,Gph,Gpp,andH,andthe`=3harmonicsofH(1)andH(2).TheonlycomponentofFs,s0whichdoesnotcontributetomsisFa?.Thishappensbecause^zisoff-diagonalinthechiralbasis,andhenceFa?dropsoutfromthediagonalpartoftheLF.Thenalresultfortheeffectivemassescanbecastintothefollowingform: ms ms=1+Xs0ps0 psF,1ss0,(2)where msm 1+sm ps.(2)NoticethattheLuttingertheorem[ 59 ]guaranteesonlythatthetotalareaoftheFS,(p2++p2)]TJ /F5 11.955 Tf 7.08 2.96 Td[(),isnotrenormalizedbytheinteractionbutsaysnothingaboutthepartialareas.Therefore,p+andp)]TJ /F1 11.955 Tf 10.4 1.79 Td[(inEqs. 2 and 2 mustbeconsideredasrenormalizedFermimomenta.ThemassmsinEq. 2 hasthesameformastheeffectivemassoffreeRashbafermionsmfs=pfs=@pfs=m=(1+sm=pfs),exceptfortheFermimomentaofafreesystempfsarenowreplacedbytherenormalizedFermimomentaps.FromEqs. 2 and 2 ,weseethataRashbaFLliquidisdifferentfromanSU2Soneinthattheeffectivemass(and,aswillseelater,otherFLquantities)isnotdeterminedonlybytheLandauparameters:onealsoneedstoknowtherenormalizedFermimomenta.Inthisrespect,aRashbaFLissimilartoothertwo-componentFLs[ 60 61 ].DuetohiddensymmetryoftheRashbaHamiltonianconservationofthesquareofthevelocity[ 62 ]thegroupvelocitiesoffreeRashbafermionswithoppositechiralitiesarethesame,cf.Eq. 2 .(However,themassesaredifferentbecauseofthedifferenceintheFermimomenta).Itisreasonabletoexpectthattheeeinteractionbreakshiddensymmetryandleadstosplittingofthesubbands'velocities.However,nosuchsplittingoccurstorstorderintheSOI[ 48 ],eveniftheeeinteractionistreatedtoanarbitraryorder[ 55 ].Inwhatfollows,wepresentamicroscopiccalculationoftheeffectivemasses 25

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andshowthatthevelocitiesareindeedsplitevenbyaweakeeinteractionbutonlyatlargervaluesoftheSOI.WeadoptasimplemodelofascreenedCoulombpotentialinthehigh-densitylimit(rs1).Theleading-orderresultformassrenormalizationm=m)]TJ /F5 11.955 Tf 12.91 0 Td[(1/rslnrsisobtainedalreadyforastaticscreenedCoulombpotentialU(q)=2e2=(q+pTF),wherepTF=p 2rspFistheThomas-Fermiscreeningmomentum.ThesameformofU(q)isvalidalsointhepresenceoftheSOIbecausethetotaldensityofstatesremainsthesameaslongasbothRashbasubbandsareoccupied.Theself-energyofthesubbandsisgivenby s(p,!)=)]TJ /F5 11.955 Tf 20.46 8.09 Td[(1 2Xs0Z(dq)Zd 2U(q) (2) [1+ss0cos(p+q)]TJ /F3 11.955 Tf 11.96 0 Td[(p)]gs0(p+q,!+),where gs(p,!)=1 i!)]TJ /F3 11.955 Tf 11.96 0 Td[(p,fs+(2)isthefreeGreen'sfunctioninthechiralbasiswithp,fsgivenbyEq. 2 .Atsmallrs,typicalmomentumtransfersaresmall:qpTFpF;therefore,thedifferencebetweenp+qandpcanbeneglectedandintersubbandscattering(s=)]TJ /F7 11.955 Tf 9.3 0 Td[(s0)dropsout.SimplifyingEq. 2 inthewayspeciedaboveandsubtractingthevalueoftheself-energyattheFS,weobtain s=s(p,!))]TJ /F5 11.955 Tf 11.95 0 Td[(s(ps,0) (2) =)]TJ /F11 11.955 Tf 11.29 11.36 Td[(Xq,U(q)[gs(p+q,!+))]TJ /F7 11.955 Tf 11.96 0 Td[(gs(ps+q,)].Integrationovergives(forps>0) s=Z)]TJ /F22 5.978 Tf 10.11 4.46 Td[(ps vFq
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ExpandingEq. 2 tolinearorderinpsandsolvingtheresultingintegraloverqtologarithmicaccuracy,wend s=ps vFe2lnps pTF.(2)wherethesubbandFermimomentum,pf,waschosenastheuppercutoff.Accordingly,theeffectivemassisrenormalizedas m m=1)]TJ /F7 11.955 Tf 17.1 8.09 Td[(e2 vFlnpf pTF.(2)Thisresultisthesameasfora2DelectrongaswithoutSOI,exceptthattheFermimomentum,enteringthelogarithmicfactor,isspecicforagivenband.ThisisalreadyenoughtoproducesplittingoftheFermivelocities v+)]TJ /F7 11.955 Tf 11.95 0 Td[(v)]TJ /F5 11.955 Tf 10.41 1.79 Td[(=pf+ m+)]TJ /F7 11.955 Tf 14.9 8.45 Td[(pf)]TJ ET q .478 w 225.74 -266.48 m 242.49 -266.48 l S Q BT /F7 11.955 Tf 225.74 -277.67 Td[(m)]TJ /F5 11.955 Tf 11.6 10.81 Td[(=)]TJ /F7 11.955 Tf 9.29 0 Td[(v0e2 vFlnp)]TJ ET q .478 w 314.94 -266.48 m 328.2 -266.48 l S Q BT /F7 11.955 Tf 314.94 -277.67 Td[(p+.(2)ThemechanismforthissplittingissimilartoOverhauser-typesplittingofeffectivemassesinapartiallyspin-polarizedmetal[ 63 65 ].TheresultinEq. 2 iscorrectonlytologarithmicaccuracy,whenp+p)]TJ /F1 11.955 Tf 7.09 1.79 Td[(,whichimpliesstrongSOI.InAppendix A ,wecomputetheself-energybeyondthelogarithmicaccuracyandshowthatEq. 2 isreproducedasaleadingterm.Also,wepresentthereasimplewaytogenerateanexpansionoftheself-energyin,whichconrmspreviousresultsforvelocitysplitting[ 48 66 ]. 2.4ThermodynamicPropertiesInprinciple,Eq. 2 enablesonetocomputeallthermodynamicpropertiesofachiralFL.Inthissection,weillustratehowthisprogramcanbecarriedoutinthechargesectorbycalculatingtheisothermalcompressibility(Sec. 2.4.1 )andderivingPomeranchukstabilityconditions(Sec. 2.4.2 ). 27

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2.4.1CompressibilityTheT=0compressibilityofaFLisgivenby =1 N2@N @N,(2)whereN=PsR(dp)nps=PsNsisthenumberdensityandNisthetotalparticlenumber.ThecompressibilityofachiralFLisobtainedbyasimplegeneralizationoftheoriginalLandau'sargument[ 34 ].IftheexactdispersionsofRashbasubbandsare+(p)="p++and)]TJ /F5 11.955 Tf 7.08 1.79 Td[((p)="p)]TJ /F5 11.955 Tf 10.06 2.61 Td[(+,thechemicalpotentialisdenedas=+(p+)=)]TJ /F5 11.955 Tf 7.08 1.8 Td[((p)]TJ /F5 11.955 Tf 7.08 1.8 Td[().Thevariationofthechemicalpotentialconsistsoftwoparts:therstoneisduetoachangeintheFermimomentaineachofthetwosubbandsandthesecondoneisduetoavariationofthequasiparticles'dispersions.Sincethevariationsofthechemicalpotentialarethesameforbothsubbands,wehave =@+(p) @pp++"p+=@)]TJ /F5 11.955 Tf 7.08 1.8 Td[((p) @pp)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+"p)]TJ /F5 11.955 Tf 7.09 2.61 Td[(,(2)where"paregivenbyEq. 2 ,@s(p) @p=@"ps @ps=vs,andps=(2=ps)Ns.Assumingthatthevariationsofnpinthep-spacearelocalizednearthecorrespondingFermimomenta,weintegratenpoverthemagnitudeofthemomentatoobtain"ps=Ps0F,0ss0Ns0=s.CombiningEq. 2 withtheconstraintN=N++N)]TJ /F1 11.955 Tf 7.08 1.79 Td[(,wesolvetheresultingsystemforNintermsofandNtoobtain =1 N2++)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(F,0\000)]TJ /F7 11.955 Tf 11.96 0 Td[(F,0)]TJ /F6 7.97 Tf 6.59 0 Td[(++)]TJ /F11 11.955 Tf 9.07 11.47 Td[()]TJ /F7 11.955 Tf 5.48 -9.68 Td[(F,0++)]TJ /F7 11.955 Tf 11.96 0 Td[(F,0+)]TJ /F11 11.955 Tf 7.08 12.4 Td[( )]TJ /F5 11.955 Tf 5.47 -9.68 Td[(1+F,0++)]TJ /F5 11.955 Tf 12.95 -9.68 Td[(1+F,0\000)]TJ /F7 11.955 Tf 11.95 0 Td[(F,0+)]TJ /F6 7.97 Tf 7.09 9.8 Td[(2(2)with=Pss.TheresultfortheSU2Scaseisrecoveredinthelimitwhentheintra-andintersubbandcomponentsofF,0ss0becomethesame.IndeedsubsititutingF,0++=F,0\000=F,0+)]TJ /F5 11.955 Tf 10.41 2.72 Td[(=F,0)]TJ /F6 7.97 Tf 6.58 0 Td[(+=Fs,0=2intoEq. 2 ,weobtain==(1+Fs,0)N2.NoticethatEq. 2 isdifferentfromthecompressibilityofatwo-componentFL[ 67 ]becauseoftheadditionalassumptionusedinRef.[ 67 ],namely,thatnotonlythetotal 28

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numberofparticlesbutalsothenumbersofparticlesofagiventyperemainsconstantundercompression.Whilethisassumptionisjustiedinthecontextofadegenerateelectron-protonplasmaconsideredinRef.[ 67 ],itisnotapplicabletoourcasewhenparticlesofoppositechiralitiescanbeexchangedbetweentheRashbasubbands,andthusthenumberofparticleswithgivenchiralityisnotconserved. 2.4.2StabilityconditionsToobtainstabilityconditionsinthechargesector,wemustrequirethatthefreeenergybeaminimumwithrespecttoarbitrarydeformationsoftheFSswhichdonotaffectingtheirspinstructure[ 68 ]: ps())]TJ /F7 11.955 Tf 11.95 0 Td[(ps=1Xn=s,nein.(2)Correspondingvariationsintheoccupationnumberscanbewrittenas ^ns,p=(ps())]TJ /F7 11.955 Tf 11.95 0 Td[(p))]TJ /F5 11.955 Tf 11.95 0 Td[((ps)]TJ /F7 11.955 Tf 11.95 0 Td[(p),(2)Achangeinthefreeenergy =TrZ^p^np(dp)+1 2TrTr0Z^f(p,p0)^np^np0(dp)(dp0)(2)mustbepositivedenitewithrespecttosuchvariations.Using^"ss0=ss0ps ms(p)]TJ /F7 11.955 Tf 12.17 0 Td[(ps),theequationfordissimpliedto =Xs,np2s 4ms2s,n+Xs,s0,npsps0 4ms^F,nss0s,ns0,)]TJ /F4 7.97 Tf 6.58 0 Td[(n.(2)Wethusarriveatthefollowingstabilityconditionsinthechargesector: 1+F,nss>0, (2a) (1+F,n++)(1+F,n\000)>(F,ns,)]TJ /F4 7.97 Tf 6.59 0 Td[(s)2. (2b) 29

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Condition 2a isthesameasforthesingle-componentcase,whilecondition 2b indicatesthatthetwo-componentFLisstableonlyiftheinter-bandinteractionissufcientlyweak[ 67 ]. 2.4.3SpinsusceptibilityofachiralFermiliquidAsitwasmentionedinSec. 2.1 ,thesubtletiesofanonSU2SFLareallinthespinsector.Toillustratethispointmoreclearly,weproceedwithevaluatingthespinsusceptibility.InordertodothiswithintheframeworkofFLtheory,oneneedstoproperlyndthechangeintheoccupationnumberofquasiparticlesduetoanexternalmagneticeld.ConsideranexternalmagneticeldintheezdirectionandofmagnitudemuchsmallerthantheeffectiveSOeld gBHpF,(2)wheregiseffectiveg-factor.(Weneglectherethediamagneticresponseofelectrons.)Intheabsenceofanexternaleld,quasiparticlesoccupychiralstatesjp,si(withs=1)lleduptotheFermimomentaps.Thepresenceofamagneticeldaffectsthespinstructureofquasiparticlestates.Supposethatthestatesinthepresenceoftheeldarejp,hi(withh=1)lleduptotheFermimomenta~ph.(Todistinguishbetweenthequantitiesintheabsenceandinthepresenceoftheeld,wedenotethelatterwithatildeoveracorrespondingsymbol.)Theenergyfunctionalofaquasiparticleintheabsenceoftheeldisdiagonalinthejp,sibasiswitheigenvalues": "pss0=hp,sj^"pjp,s0i=ss0"ps(2)Theoccupationnumberintheabsenceoftheeldisalsodiagonal npss0=hp,sj^npjp,s0i=ss0nps,(2) 30

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wherenps=(ps)]TJ /F7 11.955 Tf 11.98 0 Td[(p).Inthepresenceoftheeld,thejp,sibasisisnotaneigenstateoftheHamiltonian.Therefore,theZeemanpartoftheenergyfunctionalisnotdiagonalinthisbasis:1 ~"pss0="pss0+"pss0,(2)where "pss0=1 2g(p)BHzss0,(2)g(p)istherenormalizedg-factorwhichdependsontheelectronmomentum,andzinthechiralbasisisgivenbythelastformulainEq. 2 .Boththeenergyfunctionalofquasiparticlesinthepresenceoftheeldandtheiroccupationnumberarediagonalinthejp,hibasis ~"phh0=hp,hj^"pjp,h0i=hh0~"ph~nphh0=hp,hj^npjp,h0i=hh0~nph (2) Thereexistsaunitarymatrix^Uthatdiagonalizes~"pss0 ~"phh0=Uyhs~"pss0Us0.(2)or,equivalently, ~"pss0=Uysh~"phh0Uh0.(2)TorstorderinH, ^U=1+H^M+O(H2),(2) 1IntheSU2Scase,thechoiceofthespinquantizationaxisisarbitrary,andtheZeemanenergyofaquasiparticlecanalwaysbewritteninthediagonalformaszH.Inthechiralcase,thechoiceofthespinquantizationaxisisunique,andtheZeemanenergycannotalwaysbereducedtodiagonalform. 31

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and^Misananti-Hermitianmatrixparameterizedas ^M=0B@ia)]TJ /F3 11.955 Tf 9.29 0 Td[(ib1CA (2) withrealaandb.Thematrix^Misdeterminedfromtheconditionthatthelinear-in-Hpartof~"pss0isgivenbyEq. 2 .Foradiagonal~"phh0,thediagonalelementsofUysh~"phh0Uh0areequaltozero,whileEq. 2 containsonlyoff-diagonalelements.ThisxesinEq. 2 toberealandequalto =g(p)B 21 "p+)]TJ /F3 11.955 Tf 11.96 0 Td[("p)]TJ /F5 11.955 Tf 8.28 10.81 Td[(.(2)Thediagonalcomponentsof^MremainundenedtorstorderinHbut,usingthat^U=eH^M,theyarefoundtobezerotoallhigherordersaswell.Consequently,^Mcanbewrittenas ^M=0B@01)]TJ /F5 11.955 Tf 9.3 0 Td[(101CA.(2)Sincethesamematrix^Udiagonalizesalsotheoccupationnumbermatrix,wehave ~npss0=Uysh~nphh0Uh0.(2)Tondalinear-in-Hcorrectionto~npss0,itsufcestoapproximate~nphh0asdiag(np+,np)]TJ /F5 11.955 Tf 7.09 2.44 Td[()witheld-independentnp.Then ~npss0=h^Mydiag(np+,np)]TJ /F5 11.955 Tf 7.09 2.44 Td[()+diag(np+,np)]TJ /F5 11.955 Tf 7.09 2.44 Td[()^Mi=H(np+)]TJ /F7 11.955 Tf 11.95 0 Td[(np)]TJ /F5 11.955 Tf 7.08 2.61 Td[()^zss0. (2) ItisatthispointwherethemaindifferencebetweentheSU2SandchiralFLsoccurs:fortheformer,thechangeintheoccupationnumberislocalizedneartheFS;forthelatter,itisproportionaltoadifferenceoftheoccupationnumbersintheabsenceoftheeld,andisthusniteforallmomentainbetweentheFSsofchiralsubbands. 32

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Withthisremarkinmind,westillproceedwithaderivationoftheequationfortherenormalizedg-factorbydecomposingtheenergyvariationintotheZeemanpartandthepartduetoavariationintheoccupationnumbers: ~"pss0=1 2g(p)BHzss0=1 2gBHzss0+Xtt0Z(dp0)fst,s0(p,p0)~np0t. (2) Since~np0tisproportionaltozt,thesumovertandt0selectstheonlycomponentoftheLFinEq. 2 thatcontainsz,i.e.,fa?.Therenormalizedg-factorremainsisotropicinthemomentumspaceuntilaPomeranchukinstabilityisreached:g(p)=g(p).Withthissimplication,Eq. 2 reducesto g(p) g=1+4Z(dp0)fa?(p,p0)(np0+)]TJ /F7 11.955 Tf 11.96 0 Td[(np0)]TJ /F5 11.955 Tf 7.86 2.61 Td[() Bg=1)]TJ /F5 11.955 Tf 11.96 0 Td[(2Zp)]TJ /F4 7.97 Tf -9.68 -22.84 Td[(p+dp0p0 2fa?,0(p,p0) "p0+)]TJ /F3 11.955 Tf 11.96 0 Td[("p0)]TJ /F7 11.955 Tf 17.49 20.42 Td[(g(p0) g, (2) wherefa?,0(p,p0)isthe`=0angularharmonicoffa?(weremindthereaderthat"pdependonlyonthemagnitudeofp).IncontrasttotheSU2Scase,theequationforg(p)remainsanintegralone,eveniftheexternalmomentumisprojectedontooneoftheFSs.Theout-of-planespinsusceptibilityisthenfoundas zz=gB 2HXss0Z(dp)zss0~ns0s=g2B 2Zp)]TJ /F4 7.97 Tf -9.68 -22.83 Td[(p+pdp 2g(p) "p+)]TJ /F3 11.955 Tf 11.96 0 Td[("p)]TJ /F5 11.955 Tf -259.84 -33.04 Td[(=g2B 224Zp)]TJ /F4 7.97 Tf -9.68 -22.84 Td[(p+dpp 21 "p+)]TJ /F3 11.955 Tf 11.96 0 Td[("p)]TJ /F2 11.955 Tf 10.93 10.81 Td[()]TJ /F5 11.955 Tf 11.96 0 Td[(2Zp)]TJ /F4 7.97 Tf -9.68 -22.84 Td[(p+dpp 2Zp)]TJ /F4 7.97 Tf -9.69 -22.84 Td[(p+dp0p0 2fa?,0(p,p0) ("p+)]TJ /F3 11.955 Tf 11.96 0 Td[("p)]TJ /F5 11.955 Tf 7.09 2.44 Td[()"p0+)]TJ /F3 11.955 Tf 11.96 0 Td[("p0)]TJ /F11 11.955 Tf 7.55 15.88 Td[(35. (2) Inthelastline,weusedEq. 2 forg(p).Equations 2 and 2 explicitlydemonstratethattheout-of-planespinsusceptibilityisdeterminedentirelybythestates 33

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inbetweenthetwospin-splitFSs.Thisisalreadytrueforanon-interactingsystemwhenthesecondterminEq. 2 vanisheswhiletherstonegivesacorrectresultfortheout-of-planesusceptibilityoftheRashbaFermigaswithbothsubbandsoccupied,i.e.,fzz=g2Bm=4(cf.Ref.[ 44 ]).(InAppendix B ,weshowexplictlythatthecontributionfromstatesneartheFSsofaRashbaFermigastofzzvanishes.)ToapplyEq. 2 fortheFLcase,oneneedstoknow(renormalized)dispersions,"p,andthefa,?componentoftheLFintheentireintervalofmomentainbetweentheFSs.IftheSOcouplingisnotweak,onethusneedstoknowthepropertiesofchiralQPsfarawayfromtheirrespectiveFSs,whichisoutsidethescopeoftheFLtheory.ThisproblemisnotmerelytechnicalbutfundamentalbecauseQPsdecayawayfromtheirFSs,andonethuscannotformulatetheFLtheoryasatheoryofwell-denedQPs.Thisdoesnotmeanthatachiralelectronsystemisanon-FL;onthecontrary,allmicroscopiccalculationsaswellasexperimentalevidencepointattheFL-natureofchiralelectronsystems.However,theyareFLswiththespinsectorthatcannotbedescribedwithintheFLtheory.ThisconclusionisnotrestrictedtothecaseofaFLwithRashbaSOIbutisalsotrueforanynon-SU2SFL,includingapartiallyspin-polarizedFL,e.g.,He3inamagneticeld[ 57 ].Nevertheless,weshowinSec. 2.5.4 thatEq. 2 reproducescorrectlybothlimitingcasesofaweakeeinteractionandaweakSOcoupling.Asaconcludingremarkforthissection,thein-planespinsusceptibility(xx=yy)canbeshowntocontainbothon-andoff-theFScontributions.However,sincethereisalwaysanitecontributionfromthedampedstatesinbetweenthetwoFSs,theproblemremainsthesameasforzz.Inaddition,thein-planespinsusceptibilitycontainshigherangularharmonicsofallLandauparametersexceptfa?. 34

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2.5Zero-SoundCollectiveModesandPerturbationTheoryfortheLandauFunctionThepurposeofthissectionistoestablishtheconnectionbetweentheLandaufunctionandinteractionverticesofthemicroscopictheory,andalsotoconrmtheformoftheLandaufunctioninEq. 2 usingperturbationtheory. 2.5.1RelationbetweentheLandaufunctionandmicroscopicverticesAsinthetheoryofaSU2SFL,therelationbetweentheLandaufunctionandmicroscopicinteractionverticesisestablishedbyderivingtheequationsofmotionforzero-soundmodes.Withnolossinthegeneralityoftheconcludingstatementsofthesection,weignoretheeffectofimpuritiesandcomplicationsarisingfromtheelectricchargeoftheelectrons.TostudythecollectivemodeswithintheframeworkofaphenomenologicalFLtheory,westartwiththe(collisionless)quantumBoltzmannequationforthenon-equilibriumpartoftheoccupationnumbermatrix: @t^np+i[^n,^"p)]TJ /F5 11.955 Tf 9.74 2.61 Td[(+vrr^np)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2rr^"p,@p^n0p+=0,(2)whereh^A,^Bidenotesthe(anti)commutatorof^Aand^B.(Forbrevity,thedependencesof^npand^"ponrandtarenotdisplayed.)Wewillbeinterestedinzero-soundmodesinthechargesectorwhichcorrespondtovariationsinthediagonalelementsof^n.Inthechiralbasis, nps(r,t)npss(r,t)=("ps)]TJ /F3 11.955 Tf 11.96 0 Td[()as(p)ei(qr)]TJ /F6 7.97 Tf 6.59 0 Td[(t),(2)whereas(p)describestheangulardependenceofnp.Hence,Eq. 2 reducesto @tnps+vsrrnps+("ps)]TJ /F3 11.955 Tf 11.96 0 Td[()vsrr"ps=0,(2)where rr"ps=Xs0Zfs,s0(p,p0)rrnp0s0(dp0).(2) 35

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Figure2-1. TheDysonequationforthescatteringvertex.Thersttermontherightsiderepresentstheregularvertex,)]TJ /F6 7.97 Tf 6.78 4.34 Td[(. UsingEq. 2 ,weobtain As=Xs0ZFss0s0As0d0 2,(2)where As=as)]TJ /F6 7.97 Tf 6.58 0 Td[(1s(2)and s=vsq )]TJ /F14 11.955 Tf 11.96 0 Td[(vsq.(2)Next,wederiveEq. 2 fromthemicroscopictheory.UsingtheDysonequationfortheinteractionvertexseeFig. 2-1 wearriveat )]TJ /F4 7.97 Tf 6.78 -1.79 Td[(s,r;s0,r0(P,K;Q)=)]TJ /F6 7.97 Tf 27.6 4.94 Td[(s,r;s0,r0(P,K) (2) +ZP0)]TJ /F6 7.97 Tf 6.77 4.94 Td[(s,t;s0,t0(P,P0)t,t0(P0;Q))]TJ /F4 7.97 Tf 11.66 -1.79 Td[(t0,r;t,r0(P0,K;Q),wheres...t0=1labeltheRashbasubbands,the2+1momentaaredenedasP=(!,p),P0=(!0,p),etc.,andRPisashorthandnotationfor(2))]TJ /F6 7.97 Tf 6.58 0 Td[(3Rd!Rd2p....Furthermore,)]TJ /F1 11.955 Tf 10.1 0 Td[(istheexactvertexwhichcontainsthepolescorrespondingtothecollectivemodes,)]TJ /F6 7.97 Tf 6.77 4.34 Td[(isaregularvertex,obtainedfrom)]TJ /F1 11.955 Tf 10.1 0 Td[(inthelimitofq=!0and!0,[ 34 ]andss0istheparticle-holecorrelator.Theoff-diagonalcomponentsofss0aregappedbecauseofspin-orbitsplittingoftheRashbasubbands,whilediagonalonescontainsingularpartsgivenby ss(P,Q)=(2iZ2s=vs)(!)(p)]TJ /F7 11.955 Tf 11.95 0 Td[(ps)s,(2) 36

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whereZsistheZ-factorofsubbands.TheLandaufunctionisrelatedtothevertex)]TJ /F6 7.97 Tf 6.77 4.34 Td[(s,r;s,r.SinceEq. 2 mustholdforanyK,vertexcanbewrittenastheproductoftwoindependentcontributions: )]TJ /F4 7.97 Tf 6.78 -1.79 Td[(s,r;s,r(P,K;Q)=ss(P;Q)rr(K;Q).(2)Nearthepolesof)]TJ /F1 11.955 Tf 6.78 0 Td[(,wehave ss=XtZ2ttZ)]TJ /F6 7.97 Tf 6.77 4.93 Td[(s,t;s,t(,0)ttd0 2,(2)ComparingthekernelsinEqs. 2 and 2 ,weidentify ^fss0=s0 sZ2s0)]TJ /F6 7.97 Tf 6.78 4.94 Td[(ss0.(2)Weseethat,exceptforthenormalizationfactor,therelationbetweenthevertexandtheLandaufunctionisthesameasintheSU2Stheory. 2.5.2LandaufunctionfromperturbationtheoryNext,weshowhowthevariouscomponentsofthephenomenologicalLFinEq. 2 arereproducedbyperturbationtheoryfortheinteractionvertex.Inthissection,weconsiderasecondorderperturbationtheoryinanite-rangeinteractionUq.InSec. 2.5.3 ,wederivesomeparticularresultsforthecaseofapoint-likeinteraction,Uq=U.AccordingtoEq. 2 ,theLFisproportionaltoacertainlimitofthe(antisymmetrized)interactionvertex)]TJ /F6 7.97 Tf 6.77 4.34 Td[(.Inthespinbasis, )]TJ /F6 7.97 Tf 6.77 4.94 Td[(,;,(p,p0)=lim!0limq !0)]TJ /F10 7.97 Tf 6.78 -1.79 Td[(,;,(P,P0;Q)j!=!0=0.(2)Torstorderintheinteraction,wehave )]TJ /F6 7.97 Tf 6.78 4.94 Td[(,;,=U0)]TJ /F7 11.955 Tf 13.15 8.52 Td[(Ujp)]TJ /F6 7.97 Tf 6.58 0 Td[(p0j 2)]TJ /F7 11.955 Tf 13.15 8.52 Td[(Ujp)]TJ /F6 7.97 Tf 6.58 0 Td[(p0j 2^^, (2) 37

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Figure2-2. Allsecond-orderdiagramsforthe)]TJ /F6 7.97 Tf 6.78 4.34 Td[(vertexrelatedtotheLandaufunction.Therestofthesecond-orderdiagramsfor)]TJ /F1 11.955 Tf 10.1 0 Td[(vanishintheq=!0limit.Theinternal2+1-momentumL0isgivenbyL0=L+P0)]TJ /F7 11.955 Tf 11.96 0 Td[(P. whichisthesameasintheSU2Scase.Toseethenon-SU2Sterms,oneneedstogotoatleastsecondorder.Westartwithaparticle-particlechanneldiagraminFig. 2-2 a,thedirect(a1)andexchange(a2)partsofwhicharegivenby )]TJ /F6 7.97 Tf 6.77 5 Td[(,a1,;,(p,p0)=)]TJ /F11 11.955 Tf 11.29 16.27 Td[(ZLU2jp)]TJ /F6 7.97 Tf 6.59 0 Td[(ljGf(L)Gf(L0) (2a) )]TJ /F6 7.97 Tf 6.77 5.01 Td[(,a2,;,(p,p0)=)]TJ /F11 11.955 Tf 11.29 16.27 Td[(ZLUjp)]TJ /F6 7.97 Tf 6.59 0 Td[(ljUjp0)]TJ /F6 7.97 Tf 6.58 0 Td[(ljGf(L)Gf(L0), (2b) 38

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correspondingly.Here,integrationgoesoverL=(l,!),L0=)]TJ /F7 11.955 Tf 9.3 0 Td[(L+KwithK=(0,k)=(0,p+p0),andthefreeGreen'sfunctioninthespinbasisreads ^Gf(P)=Xs1 2(1+s^epez)gs(P)(2)wheregs(P)isthesameasinEq. 2 .First,wefocusonthedirectterm,a1.Thecrossproductoftheunitymatricesobviouslyonlyrenormalizesthe11terminthetheLFofSU2SFL.ConsidernowtermsinvolvingoneunitymatrixandonePaulimatrix: )]TJ /F5 11.955 Tf 5.48 -9.69 Td[()]TJ /F6 7.97 Tf 6.78 5.01 Td[(,a1,;,g=)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 4[1^Xs,s0s0A0ss0(k)ez+1^Xs,s0sAss0(k)ez], (2) where Ass0(k)=ZLlU2jp)]TJ /F6 7.97 Tf 6.59 0 Td[(ljgfs(L)gfs0(L0), (2a) A0ss0(k)=ZLl0U2jp)]TJ /F6 7.97 Tf 6.58 0 Td[(ljgfs(L)gfs0(L0). (2b) SincetheGreen'sfunctiondependsonlyonthemagnitudeoftheelectronmomentum,thevectorsinEq. 2a and 2b arerelatedtoeachotherby Ass0(k)=A0s(k).(2)Forthesamereason,thedirectionsofbothvectorsmustcoincidewiththatofk.Usingthesetwoproperties,weobtain Xs,s0sAss0=Xs,s0s0A0ss0=kA1(k),(2) 39

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whereB1(k)issomescalarfunctionofk,thepreciseformofwhichdependsonthechoiceofUq.Usingthelastresult,wenallyarriveat )]TJ /F5 11.955 Tf 5.48 -9.69 Td[()]TJ /F6 7.97 Tf 6.77 5.01 Td[(,a1,;,g=)]TJ /F5 11.955 Tf 10.5 8.09 Td[(1 4B1(jp+p0j)[1^+1^](p+p0)ez[1^+1^]C1, (2) whichcoincideswiththetensorformofthegppterminEq. 2 .Likewise,exchangediagrama2gives )]TJ /F5 11.955 Tf 5.48 -9.68 Td[()]TJ /F6 7.97 Tf 6.77 5 Td[(,a2,;,g=)]TJ /F5 11.955 Tf 10.5 8.08 Td[(1 4A2(jp+p0j)[1^+1^](p+p0)ez[1^+1^]C2, (2) whereA2isdenedbythesamerelationsas 2a and 2 ,exceptforareplacement U2jp)]TJ /F6 7.97 Tf 6.59 0 Td[(lj!Ujp)]TJ /F6 7.97 Tf 6.58 0 Td[(ljUjp0)]TJ /F6 7.97 Tf 6.59 0 Td[(lj.(2)Althoughthetensorformoftheexchangevertexseemstobedifferentfromthatofthedirectone,itis,infact,thesame.Toseethis,itisconvenienttotransformthevertexintothechiralbasisusing )]TJ /F6 7.97 Tf 6.78 4.93 Td[(ss0(p,p0)=X)]TJ /F6 7.97 Tf 6.77 4.93 Td[(,;,(p,p0)hjs,pihjs0,p0ihs,pjihs0,p0ji, (2) 40

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wherejs,kiistheRashbaspinordenedbyEq. 2 .Applying 2 toEqs. 2 and 2 ,weobtainforthedirectandexchangecontributions,correspondingly )]TJ /F6 7.97 Tf 6.78 5.01 Td[(,a1ss0=(hs,pj^js,pi+hs0,p0j^js0,p0i)C1 (2a) )]TJ /F6 7.97 Tf 6.78 5 Td[(,a2ss0=1 2h1+ssi(p)]TJ /F10 7.97 Tf 6.59 0 Td[(p0)ihs,pj^js0,p0iC2+1 2h1+ssi(0p)]TJ /F10 7.97 Tf 6.59 0 Td[(p)ihs0,p0j^js,piC2. (2b) CalculatingthematrixelementsofthePaulimatrices,itcanbereadilyshownthatthevectorstotheleftfromC1in 2a andfromC2in 2b arethesame.Next,weconsidertheterminvolvingtwoPaulimatricesinEq. 2a )]TJ /F5 11.955 Tf 5.48 -9.68 Td[()]TJ /F6 7.97 Tf 6.77 5 Td[(,a1,;,h=)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 4(ez^)i(ez^)jS(1)ij(p,p0), (2) whereS(1)ijisasecondranktensor S(1)ij(p,p0)=Xs,s0ss0ZL^li^l2jjp)]TJ /F6 7.97 Tf 6.59 0 Td[(ljgfs(L)gfs0(L0).(2)Thistensordependsonlyonk=p+p0,andcanbeformedonlyfromofthecomponentsofkas S(1)ij(k)=kikjS(1)(k).(2)Hence )]TJ /F5 11.955 Tf 5.48 -9.68 Td[()]TJ /F6 7.97 Tf 6.77 5 Td[(,a1,;,h=)]TJ /F5 11.955 Tf 10.49 8.08 Td[(1 4S(1)(jp+p0j)[^(p+p0)ez)][^(p+p0)ez], (2) whichreproducesthetensorstructureofthehtermsinEq. 2 (withh(1)=h(2)=h=2tothisorderoftheperturbationtheory).Asbefore,theexchangediagramproducessametypeofterms.Sincethetensorstructuresofthedirectandexchangeparticle-particle 41

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diagramsarethesame,thenon-SU2Spartofthetotalparticle-particlevertex)]TJ /F6 7.97 Tf 6.77 4.34 Td[(,a=)]TJ /F6 7.97 Tf 6.77 4.34 Td[(,a1)]TJ /F5 11.955 Tf 12.2 0 Td[()]TJ /F6 7.97 Tf 6.78 4.34 Td[(,a2vanishesforapoint-likeinteraction,Uq=U.Therefore,thegpptermintheLFisabsentinthiscase,whereasthehtermscomefromonlytheparticle-holechannel,consideredbelow.Next,weshowthatthecrosseddiagramintheparticle-holechannel(Fig. 2-2 d)producesboththegphandhtermsinEq. 2 .TheterminvolvingoneunitymatrixandonePaulimatrixresultingfromthedirectdiagramisprocessedinthesamewayastheanalogoustermfromtheparticle-particlediagram.TheonlydifferenceisthatvectorsAss0andA0ss0arenowreplacedby Bss0(k)=ZLlU2jp)]TJ /F6 7.97 Tf 6.59 0 Td[(ljgfs(L)gfs0(L00), (2a) B0ss0(k)=ZLl0U2jp)]TJ /F6 7.97 Tf 6.58 0 Td[(ljgfs(L)gfs0(L00), (2b) whereL00=L+P0)]TJ /F7 11.955 Tf 11.95 0 Td[(P,andBandB0arerelatedby Bss0(k)=)]TJ /F14 11.955 Tf 9.3 0 Td[(B0s(k).(2)DeningascalarfunctionB(k)via Xs,s0sBss0=)]TJ /F11 11.955 Tf 11.3 11.35 Td[(Xs,s0s0B0ss0=kB(k),(2)weobtain )]TJ /F5 11.955 Tf 5.48 -9.68 Td[()]TJ /F6 7.97 Tf 6.77 5 Td[(,d,;,g=)]TJ /F5 11.955 Tf 10.49 8.08 Td[(1 4B(jp)]TJ /F14 11.955 Tf 11.95 0 Td[(p0j)[1^(p0)]TJ /F14 11.955 Tf 11.95 0 Td[(p)ez)]TJ /F5 11.955 Tf 11.96 0 Td[(1^(p0)]TJ /F14 11.955 Tf 11.95 0 Td[(p)ez], (2) 42

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whichcoincideswiththetensorformofthegphterminEq. 2 .TheterminvolvingtwoPaulimatricesiscastintoaformsimilartoEq. 2 : )]TJ /F5 11.955 Tf 5.48 -9.69 Td[()]TJ /F6 7.97 Tf 6.77 5.01 Td[(,d,;,h=)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 4T(jp)]TJ /F14 11.955 Tf 11.96 0 Td[(p0j)[^(p)]TJ /F14 11.955 Tf 11.95 0 Td[(p0)ez)][^(p)]TJ /F14 11.955 Tf 11.96 0 Td[(p0)ez], (2) whereT(k)isdenedsimilarlytoEqs. 2 and 2 exceptL0isreplacedbyL00=L+P0)]TJ /F7 11.955 Tf 12.59 0 Td[(P.Thelastexpressionreproducesagainthetensorialstructureoftheh(1,2)andhterms,exceptfornowh(1)=h(2)=)]TJ /F7 11.955 Tf 9.3 0 Td[(h=2.Theexchangecounterparttodiagramb1doesnotdependontheexternalmomenta.Thefore,thedirectandexhangepartofdiagrambdonotcanceleachotherevenforthecaseofUq=U.Therestofthesecond-orderdiagramsrenormalizesonlythetheSU2SpartoftheLF.Wepostponeanalysisofthesediagramsuntilthenextsection. 2.5.3Example:massrenormalizationInthissection,weapplytheformalismdevelopedinprevioussectionstoamicroscopiccalculationofmassrenormalizationforRashbafermions.Weconsideraweak,short-rangeeeinteraction(Uq=U),andassumealsothattheSOIisweakaswell,i.e.,vF.Ourfocuswillbeontheleading,O(U22)correctiontotheeffectivemassesofRashbafermions.Althoughsuchcorrectionseemstobeperfectlyanalytic,infact,itisnot.ItwillbeshownthatthecorrectiontotheeffectivemassofthesubbandscomesassU22.Sincetheeigenenergiescontainsandonlyasacombinationofs,thesecond-ordertermintheregularexpansioninsisthesamefors=1.Inthissense,thesU22correctionisnon-analytic,andwillbeshowntocomefromtheKohnanomaliesofthevariousvertices.Tosimplifynotations,wewillsuppressthesuperscriptintheverticesbecausealltheverticesconsideredinthissectionareofthe)]TJ /F6 7.97 Tf 6.78 4.34 Td[(type.Also,wewillsuppressthesuperscriptflabelingthequantitiespertinenttothefreesystem:inastraightforwardperturbationtheory,consideredhere,theinteractionshowsonlyastheU2prefactor. 43

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Anon-analyticpartoftheLFtocomesfromparticle-holediagramsfor)]TJ /F6 7.97 Tf 6.77 4.34 Td[(inFig. 2-2 ,i.e.,fromdiagramsb-d.TomaximizetheeffectoftheKohnanomaly,oneneedstoselectthoseinitialandnalstatesthatcorrespondtotheminimalnumberofsmallmatrixelementsforintra-bandbackscattering.DetailedcalculationsofthediagramsarepresentedinAppendix D ;here,weillustratehowanO(U22)correctionoccursusingdiagramcasanexample.Explicitly,thisdiagramreads )]TJ /F4 7.97 Tf 6.77 4.94 Td[(css0(ps,p0s)=)]TJ /F7 11.955 Tf 10.5 8.09 Td[(U2 2[1+ss0cos(p0)]TJ /F3 11.955 Tf 11.96 0 Td[(p)](jps)]TJ /F14 11.955 Tf 11.96 0 Td[(p0s0j),(2)where(q)isthepolarizationbubbleoffreeRashbafermions.WithouttheSOI,theKohnanomalyofislocatedatq=2pF,wherepFistheFermimomentumat=0.In2D,thepolarizationbubbleisindependentofqforq2pFandexhibitsacharacteristicsquare-rootanomalyforq>2pF.SOcouplingsplitsthespectrumintotwoRashbasubbandswithFermimomentapgivenbyEq. 2 .InRef.[ 42 ],itwasshownthatthestaticpolarizationbubble(q)remainsconstantforq2p+foranarbitraryvalueof(butaslongasbothsubbandsareoccupied).Intheregion2p+q2p)]TJ /F1 11.955 Tf 7.09 1.79 Td[(,thepolarizationbubbleexhibitsanon-analyticdependenceonq.Atsmall,allthreeFermimomentaareclosetoeachother:p+p)]TJ /F2 11.955 Tf 10.41 1.8 Td[(pF.Inthiscase,thesingularpartsofcanbewrittenas(seeAppendix C ) (q)=)]TJ /F3 11.955 Tf 9.3 0 Td[(+++(q)++)]TJ /F5 11.955 Tf 7.08 1.8 Td[((q) (2) ++(q)=)]TJ /F3 11.955 Tf 10.49 8.09 Td[( 6(q)]TJ /F5 11.955 Tf 11.95 0 Td[(2p+)q)]TJ /F5 11.955 Tf 11.96 0 Td[(2p+ pF3=2 (2) +)]TJ /F5 11.955 Tf 7.09 1.79 Td[((q)= 2(q)]TJ /F5 11.955 Tf 11.96 0 Td[(2pF)q)]TJ /F5 11.955 Tf 11.95 0 Td[(2pF pF1=2. (2) TheKohnanomaliesofaffectstheamplitudesofbackscatteringprocesseswithp0)]TJ /F14 11.955 Tf 21.92 0 Td[(p,hencep0)]TJ /F3 11.955 Tf 11.96 0 Td[(p=)]TJ /F3 11.955 Tf 11.96 0 Td[(withjj1.Therefore, )]TJ /F4 7.97 Tf 6.77 4.93 Td[(css0=)]TJ /F7 11.955 Tf 10.49 8.08 Td[(U2 21+ss0)]TJ /F7 11.955 Tf 11.95 0 Td[(ss01 22(jps)]TJ /F14 11.955 Tf 11.95 0 Td[(p0s0j),(2) 44

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TheKohnanomalyof++affecttheprocesseswith2p+q2p)]TJ /F1 11.955 Tf 7.09 1.79 Td[(.Since++alreadycontainsasmallfactorof(q)]TJ /F5 11.955 Tf 12.55 0 Td[(2p+),reectingthesmallnessofthematrixelementofintrabandbackscattering,thesingularcontributionfrom++ismaximalforinterbandprocesses(s=)]TJ /F7 11.955 Tf 9.29 0 Td[(s0),whentheangularfactorin 2 isalmostequalto1.Theonlyvertexofthistypeis )]TJ /F4 7.97 Tf 6.77 4.94 Td[(c+)]TJ /F5 11.955 Tf 10.41 2.95 Td[(=)]TJ /F7 11.955 Tf 9.29 0 Td[(U2++(q)=U2 6(q)]TJ /F5 11.955 Tf 11.96 0 Td[(2p+)q)]TJ /F5 11.955 Tf 11.96 0 Td[(2p+ pF3=2.(2)TheKohnanomalyof+)]TJ /F1 11.955 Tf 10.41 1.8 Td[(givesaneffectofthesameorderasin 2 butforintrabandprocesses: )]TJ /F4 7.97 Tf 6.77 4.93 Td[(c\000=)]TJ /F7 11.955 Tf 9.29 0 Td[(U22 2+)]TJ /F5 11.955 Tf 7.08 1.8 Td[((q). (2) Forabackscatteringprocess, q=jps)]TJ /F14 11.955 Tf 11.96 0 Td[(p0s0jps+ps0)]TJ /F7 11.955 Tf 11.95 0 Td[(pF2=4,(2)wherethedependenceoftheprefactorofthe2termoncanbeandwasneglected.For)]TJ /F4 7.97 Tf 6.77 4.34 Td[(c\000,wehaveq=2p)]TJ /F2 11.955 Tf 9.74 1.79 Td[()]TJ /F7 11.955 Tf 11.96 0 Td[(pF2=4.Expressing2intermsofq,wearriveat )]TJ /F4 7.97 Tf 6.77 4.93 Td[(c\000=)]TJ /F3 11.955 Tf 9.3 0 Td[(U2(q)]TJ /F5 11.955 Tf 11.95 0 Td[(2pF)(q)]TJ /F5 11.955 Tf 11.96 0 Td[(2p)]TJ /F5 11.955 Tf 7.08 1.63 Td[()(q)]TJ /F5 11.955 Tf 11.95 0 Td[(2pF)1=2 p3=2F. (2) Nowitisobviousthat)]TJ /F4 7.97 Tf 6.77 4.34 Td[(c+)]TJ /F1 11.955 Tf 10.41 2.96 Td[(and)]TJ /F4 7.97 Tf 6.77 4.34 Td[(c\000areofthesameorder. 45

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TheremainingdiagramsareevaluatedinAppendix D withthefollowingresults )]TJ /F4 7.97 Tf 6.78 4.94 Td[(b\000=0 (2) )]TJ /F4 7.97 Tf 6.78 4.94 Td[(b+)]TJ /F5 11.955 Tf 17.05 2.95 Td[(=)]TJ /F5 11.955 Tf 9.3 0 Td[()]TJ /F4 7.97 Tf 6.77 4.94 Td[(c+)]TJ /F5 11.955 Tf 10.41 2.95 Td[(=U2(q)]TJ /F5 11.955 Tf 11.95 0 Td[(2pF)q)]TJ /F5 11.955 Tf 11.96 0 Td[(2pF pF3=2 (2) )]TJ /F4 7.97 Tf 6.78 4.94 Td[(d\000=U2 12(q)]TJ /F5 11.955 Tf 11.95 0 Td[(2pF)q)]TJ /F5 11.955 Tf 11.96 0 Td[(2pF pF3=2 (2) )]TJ /F4 7.97 Tf 6.78 4.94 Td[(d+)]TJ /F5 11.955 Tf 17.05 2.95 Td[(=U2 24(q)]TJ /F5 11.955 Tf 11.95 0 Td[(2p+)q)]TJ /F5 11.955 Tf 11.96 0 Td[(2p+ pF3=2 (2) )]TJ /F4 7.97 Tf 6.78 4.93 Td[(b,c,d++=0. (2) Forq,onesubstitutesq=jp)]TJ /F2 11.955 Tf 9.16 1.79 Td[()]TJ /F14 11.955 Tf 11.38 0 Td[(p0)]TJ /F2 11.955 Tf 7.08 2.95 Td[(jintoEqs. 2 2 and 2 ,andq=jp+)]TJ /F14 11.955 Tf 11.38 0 Td[(p0)]TJ /F2 11.955 Tf 7.09 2.95 Td[(jintoEqs. 2 2 and 2 .Nowwecancalculatetheangularharmonicsofthevertices.Combiningtheverticesinthe\000channelandtransformingbackfromqto,weobtainforthenthharmonicofthetotalvertexinthischannel )]TJ /F6 7.97 Tf 6.77 4.93 Td[(,n\000=)]TJ /F4 7.97 Tf 19.39 4.93 Td[(c,n\000+)]TJ /F4 7.97 Tf 18.73 4.93 Td[(d,n\000=)]TJ /F5 11.955 Tf 9.3 0 Td[(()]TJ /F5 11.955 Tf 9.3 0 Td[(1)nU2 4Z2q 2m pF0d 22m pF)]TJ /F3 11.955 Tf 13.15 8.09 Td[(2 41=2+()]TJ /F5 11.955 Tf 9.3 0 Td[(1)nU2 12Z2q 2m pF0d 2m pF)]TJ /F3 11.955 Tf 13.15 8.09 Td[(2 43=2=)]TJ /F5 11.955 Tf 10.5 8.09 Td[(3 8()]TJ /F5 11.955 Tf 9.3 0 Td[(1)nU2m pF2. (2) Equation 2 isvalidfor1np pF=8m.Likewise,wendforthe+)]TJ /F1 11.955 Tf 12.62 0 Td[(channel )]TJ /F6 7.97 Tf 6.77 4.94 Td[(,n+)]TJ /F5 11.955 Tf 10.41 2.95 Td[(=)]TJ /F4 7.97 Tf 19.39 4.94 Td[(d,n+)]TJ /F5 11.955 Tf 10.4 2.95 Td[(=1 16()]TJ /F5 11.955 Tf 9.3 0 Td[(1)nU2m pF2. (2) TheharmonicsoftheLFarerelatedtothoseofverticesviaEq. 2 withtheproportionalitycoefcients0Z2s0=s,whichistakentobeequaltoonetolowestorderinUand.AsmentionedinSec. 2.3 ,theformulafortheeffectivemass,Eq. 2 ,containsnotonlythecomponentsoftheLFbutalsotheratioofthesubbandFermimomentawhich,ingeneral,arerenormalizedbytheinteraction.However,combiningpreviousresultsfromRefs.[ 45 55 66 ],onecanshowthatthiseffectoccursonlytohigherorders 46

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inandU.Indeed,ananalyticpartofthegroundstateenergyoftheelectronsystemwithRashbaSOIcanbewrittenasEan=C)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(n=n)]TJ /F5 11.955 Tf 11.95 0 Td[(2=vfF2,whereN=N)]TJ /F2 11.955 Tf 10.4 1.8 Td[()]TJ /F7 11.955 Tf 12.62 0 Td[(N+isthedifferenceinthenumberofelectronsoccupyingtheRashbasubbandsandvfFisthebareFermivelocity[ 55 ].Thisresultisvalidtoallordersintheeeinteractionofarbitrarytype,andtolowestorderin.Therefore,theminimumofEancorrespondstothesamevalueofNasforafreeelectrongas,whichmeansthattheFermimomentaarenotrenormalized.RenormalizationofNoccursbecauseofnon-analytictermsinthegroundstateenergy[ 45 66 ].Forashort-rangeinteraction,therstnon-analyticcorrectionoccurstofourthorderinU:Ena/U4jj3ln2jj.(Acubicdependenceonjjisbecauseanon-analyticpartoftheenergyin2Dscalesasthecubeoftheparametercontrollingnon-analyticity;[ 69 ]anadditionalfactorofln2jjcomesfromrenormalizationoftheinteractionintheCooperchannel.)ThentheminimumofEan+EnacorrespondstoachangeintheFermimomentap)]TJ /F2 11.955 Tf 10.27 1.79 Td[()]TJ /F7 11.955 Tf 12.49 0 Td[(p+/N/U42ln2jj,whichisbeyondtheorderoftheperturbationtheoryconsideredhere.Tolowestorderin,wecanalsotakep+tobeequaltop)]TJ /F1 11.955 Tf 7.09 1.8 Td[(.SubstitutingEqs. 2 and 2 intoEq. 2 ,weobtainfortherenormalizedmasses(mdenotesthedifferencebetweentherenormalizedandbarevalues): m+ m+=F,1+++F,1+)]TJ /F5 11.955 Tf 10.4 2.96 Td[(=)]TJ /F5 11.955 Tf 13.64 8.08 Td[(1 16Um pF2 (2) m)]TJ ET q .478 w 128.04 -462.11 m 153.15 -462.11 l S Q BT /F7 11.955 Tf 132.22 -473.3 Td[(m)]TJ /F5 11.955 Tf 22.43 9.99 Td[(=F,1\000+F,1)]TJ /F6 7.97 Tf 6.59 0 Td[(+=5 16Um pF2, (2) WeseethatmassrenormalizationisdifferentfordifferentRashbasubbands.SincetheFermimomentaarenotrenormalizedtothisorder,thisimpliesthatthegroupvelocitiesv=p=marealsodifferent.AswepointedoutinSec. 2.3 ,theequalityofthesubbands'groupvelocitiesispreservedtoanarbitraryorderintheeeinteraction,providedthattheSOIistreatedtorstorder[ 55 ].WithintheRPAforaCoulombpotential,velocitysplittingoccursbecauseofanon-analyticrsjj3lnjjtermintheelectronself-energy[ 66 ]whereasinourmodel 47

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Figure2-3. Selfenergydiagramstorstorderintheinteraction. velocitysplittingoccursalreadyatorderU22.ThedifferencebetweenourresultandthatofRef.[ 66 ]isthatourrenormalizationcomesfromtheKohnanomalywhichwasnotconsideredinRef.[ 66 ].NotethatthemodelassumptionofUq=Uusedinthissectionisnotessential:formomentum-dependentinteractionUq,onesimplyneedstoreplaceUbyU2pFinthenalresults.Inthehigh-densitylimit(rs1),the2pFcomponentofascreenedCoulombpotentialisoforderr2s;therefore,ourresultfortheCoulombcasewouldtranslateintov+)]TJ /F7 11.955 Tf 12.19 0 Td[(v)]TJ /F2 11.955 Tf 11 1.79 Td[(/r2s2.Termsoforderr2swerenotconsideredinRef.[ 66 ].Inreality,boththersjj3lnjjandr2s2termsarepresent,andthecompetitionbetweenthetwoiscontrolledbytheratiooftwosmallparameters:jjlnjj=rs. 2.5.4PerturbationtheoryforzzInthissection,weshowthatEq. 2 reproducestheknownresultsforzzinthelimitingcasesofaweakeeinteractionoraweakSOcoupling. 2.5.4.1Weakelectron-electroninteractionFirst,weexaminethelimitofaweakandshort-rangeeeinteraction(torstorderintheinteractionamplitudeUwithoutmakinganyassumptionaboutthestrengthoftheSOcoupling.)ItfollowsfromEq. 2 thatfa?=)]TJ /F7 11.955 Tf 9.3 0 Td[(U=2torstorderintheinteraction.Wealsoneedtheself-energyofthequasi-particlestorstorderintheinteractiongivenby(seeFig. 2-3 ) s(P)=as(P)+bs(P),(2) 48

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as(P)=)]TJ /F7 11.955 Tf 10.5 8.09 Td[(U 2Xs0ZK[1+ss0cos(k)]TJ /F3 11.955 Tf 11.96 0 Td[(p)gs(K)=)]TJ /F7 11.955 Tf 10.49 8.09 Td[(U 2N, (2) andsimilarlybs(P)=UN,whereNisthetotalnumberdensityofelectrons.Therefore,theself-energiesofbothRashbabranchesareconstantandequaltoeachother.Sincetheshiftintheenergyofquasiparticleswithoppositechiralitiesisthesame,wehave"p+)]TJ /F3 11.955 Tf 12.25 0 Td[("p)]TJ /F5 11.955 Tf 11.14 2.61 Td[(=2pandp)]TJ /F2 11.955 Tf 10.04 1.79 Td[()]TJ /F7 11.955 Tf 12.25 0 Td[(p+=2m,whicharethesamerelationsasforafreesystem.SubstitutingtheseresultsintoEq. 2 ,wendthattorstorderinU zz=01+mU 2. (2) Asistobeexpected,thisresultcoincideswiththatgivenbytherstladderdiagramoftheperturbationtheory[ 44 ].NoticethatzzinEq. 2 doesnotdependon.ThisfeaturesurvivestoallordersinUwithintheladderapproximation.Beyondtheladderapproximation,theleadingdependenceonoccurstoorderU2asanon-analyticcorrection:zz=(2=3)0(mU=4)2jjpF=F(Refs.[ 44 45 ]). 2.5.4.2Weakspin-orbitcouplingNext,weconsidertheoppositelimitofanarbitraryeeinteractionsbutinnitesimallysmallSOcoupling.Inthislimit,onemustrecovertheresultofanSU2SFL.Toobtainzzinthislimit,oneneedstoevaluateintegralsofthetypeZp)]TJ /F4 7.97 Tf -9.69 -22.84 Td[(p+dp0 2fa?,0(p,p0)g(p0) "p0+)]TJ /F3 11.955 Tf 11.95 0 Td[("p0)]TJ /F5 11.955 Tf 31.61 12.33 Td[(,tozerothorderin.InnitesimalSOcouplingimpliesthattheregionofintegrationisinnitesimallysmallcomparedtotheFermimomentum,anditsufcestoconsideronlythelinearpartofthedispersionneartheFS"ps=vs(p)]TJ /F7 11.955 Tf 12.18 0 Td[(ps),wherevs=@p"psjp=ps.Notethatinthepresenceoftheinteraction,ingeneral,v+6=v)]TJ /F1 11.955 Tf 7.09 1.8 Td[(.However,toobtainzztozerothorderin,onecankeeptheSOcouplingonlyintheFermimomentaofthetwosubbandsandsettozeroeverywhereelse.Asaresult,vscanbesettoequaltovF, 49

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theFermivelocityintheabsenceoftheSOcoupling.Therefore, Zp)]TJ /F4 7.97 Tf -9.68 -22.84 Td[(p+dp0 2fa,0(p,p0)g(p0) "p0+)]TJ /F3 11.955 Tf 11.95 0 Td[("p0)]TJ /F5 11.955 Tf 31.64 12.33 Td[(=pF 2vFfa?,0(pF,pF)g(pF),(2)wherefa,0(pF,pF)=fa?,0(pF,pF)j=0isthespin-asymmetriccomponentoftheLFintheabsenceoftheSOcoupling.SubstitutingthisresultintoEq. 2 andsolvingforg(pF),wereproducetheresultforanSU2SFL g(pF)=g 1+Ffa?(pF,pF),(2)whereF=pF=vFisthe(renormalized)densityofstatesattheFermilevel.Similarly, zzg22B 2Zp)]TJ /F4 7.97 Tf -9.68 -22.84 Td[(p+dppF 2g(pF) "p+)]TJ /F3 11.955 Tf 11.96 0 Td[("p)]TJ /F5 11.955 Tf 11.61 10.81 Td[(=g22B 4F 1+Ffa,0.(2)NoticethattheSOcouplingiseliminatedasananomaly,i.e.,asthecancellationbetweenasmalldenominator("p+)]TJ /F3 11.955 Tf 11.95 0 Td[("p)]TJ /F1 11.955 Tf 7.09 2.61 Td[()andanarrowintegrationrange(p)]TJ /F2 11.955 Tf 9.74 1.79 Td[()]TJ /F7 11.955 Tf 11.95 0 Td[(p+). 2.6ChiralSpinWavesInthissection,westudycollectivemodesinthespinsectorofachiralFL,andobtainthespectrumandequationsofmotionforchiralspin-wavecollectivemodes.AsdiscussedinSec. 2.4.3 ,thespinsectorofachiralFLwitharbitrarystrongSOIdoesnotadmitadescriptionintermsofwell-denedQPs.However,iftheSOIisweak,itcanbetreatedasaperturbationtotheSU2S-FL;thisapproachwasemployedinRef.[ 51 ]forstudyingtheq=0chiralspinresonances.Inwhatfollows,weextendthisapproachtothenite-qcaseandstudythedispersionrelationsintheentireqspace. 2.6.1Dispersionofchiralspin-wavesTondthespectrumofthecollectivemodes,weconsidertheDysonequationfortheinteractionvertexinEq. 2 withthekernel ss0(P,Q)=(2iZ2=vF)(!)(p)]TJ /F7 11.955 Tf 11.96 0 Td[(pF)ss0,(2) 50

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whereZistheQPrenormalizationfactorintheabsenceofSO, )]TJ /F6 7.97 Tf 6.59 0 Td[(+=+vq !)]TJ /F5 11.955 Tf 11.96 0 Td[()]TJ /F14 11.955 Tf 11.96 0 Td[(vq,+)]TJ /F5 11.955 Tf 10.4 1.79 Td[(=)]TJ /F5 11.955 Tf 9.29 0 Td[(+vq !+)]TJ /F14 11.955 Tf 11.96 0 Td[(vq,++=\000=vq !)]TJ /F14 11.955 Tf 11.95 0 Td[(vq, (2) and=2pF.Toinvestigatethecollectivemodesinthespinsector,weneedtokeeponlythespinpartof)]TJ /F6 7.97 Tf 6.78 4.34 Td[(.IntheperturbationtheorywithrespecttotheSOI,weneglectitseffectontheLFwhichistakenthesameasforanSU2SFL,Eq. 2 .Wefurtheradoptthes-waveapproximation,inwhichFa=constFa,0.Thisapproximationallowsonetoobtainaclosed-formsolutionofEq. 2 withoutaffectingtheresultsqualitatively.Oneoftheeffectsnotcapturedbythisapproximationisrenormalizationof,whichiscontrolledbyFa,1[ 38 51 ]andisthusabsentinthes-waveapproximation.Toaccountforthiseffect,onecansimplyreplacebyinthenalresults.Inthes-waveapproximation,)]TJ /F6 7.97 Tf 6.77 4.33 Td[(asexpressedas Z2F)]TJ /F6 7.97 Tf 6.77 4.94 Td[(s,t;s0(P,P0)=Fa,0hs0pjjspiht0p0jjtp0i,(2)andhs0p0jjspiarethePaulimatricesinthechiralbasis,givenbyEq. 2 .Notethateveninthes-waveapproximation,)]TJ /F6 7.97 Tf 6.77 4.34 Td[(s,t;s0dependsonthedirectionsoftheelectronmomentaviathePaulimatrices.SinceEq. 2 holdsforanyK,theexactvertexcanbefactorizedas )]TJ /F4 7.97 Tf 6.77 -1.79 Td[(s,r;s0,r0(P,K;Q)=ss0(P;Q)rr0(K;Q)(2)Nearthepolesof)]TJ /F1 11.955 Tf 6.78 0 Td[(,wehave ss0=Fa,0 2Xt,t0Zd0 2hs0pFjjspFiht0p0Fjjtp0Fitt0t.(2) 51

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Changingthevariablesasss0=sss0,weobtainasetofequationsfornormalmodes )]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F6 7.97 Tf 6.59 0 Td[(++)]TJ /F5 11.955 Tf 17.05 1.79 Td[(=Fa,0 2Zdp0 2(+)]TJ /F5 11.955 Tf 9.75 1.79 Td[(+)]TJ /F6 7.97 Tf 6.59 0 Td[(+)+Zdp0 2cos#(+)]TJ /F2 11.955 Tf 9.74 1.79 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F6 7.97 Tf 6.58 0 Td[(+)+iZdp0 2sin#(++)]TJ /F3 11.955 Tf 11.95 0 Td[(\000))]TJ /F6 7.97 Tf 6.59 0 Td[(1+)]TJ /F3 11.955 Tf 7.08 2.95 Td[()]TJ /F6 7.97 Tf 6.59 0 Td[(+=Fa,0 2Zdp0 2(+)]TJ /F5 11.955 Tf 9.75 1.79 Td[(+)]TJ /F6 7.97 Tf 6.59 0 Td[(+))]TJ /F11 11.955 Tf 11.96 16.28 Td[(Zdp0 2cos#(+)]TJ /F2 11.955 Tf 9.74 1.79 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F6 7.97 Tf 6.58 0 Td[(+))]TJ /F7 11.955 Tf 20.45 0 Td[(iZdp0 2sin#(++)]TJ /F3 11.955 Tf 11.95 0 Td[(\000))]TJ /F6 7.97 Tf 6.59 0 Td[(1++++=Fa,0 2iZdp0 2sin#(+)]TJ /F2 11.955 Tf 9.74 1.8 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F6 7.97 Tf 6.59 0 Td[(+)+Zdp0 2cos#(++)]TJ /F3 11.955 Tf 11.96 0 Td[(\000))]TJ /F6 7.97 Tf 6.59 0 Td[(1\000\000=)]TJ /F7 11.955 Tf 10.49 8.09 Td[(Fa,0 2iZdp0 2sin#(+)]TJ /F2 11.955 Tf 9.74 1.79 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F6 7.97 Tf 6.58 0 Td[(+)+Zdp0 2cos#(++)]TJ /F3 11.955 Tf 11.96 0 Td[(\000), (2) where #p0)]TJ /F3 11.955 Tf 11.95 0 Td[(p.(2)Thelasttwoequationofset 2 implythat++=)]TJ /F3 11.955 Tf 9.3 0 Td[(\000.Expandingthenormalmsodesintoangularharmonicsas ss0=1Xn=0,nss0cos(n)+1Xn=1,nss0sin(n),(2)weseethat,n++couplesonlyto,ms,)]TJ /F4 7.97 Tf 6.59 0 Td[(swhile,n++couplesonlyto,ms,)]TJ /F4 7.97 Tf 6.59 0 Td[(s.Introducingnewcoordinates X=,1+)]TJ /F2 11.955 Tf 9.74 2.72 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(,1)]TJ /F6 7.97 Tf 6.58 0 Td[(+ 2+i,1++Y=,1+)]TJ /F2 11.955 Tf 9.74 2.72 Td[()]TJ /F5 11.955 Tf 12.66 0 Td[(,1)]TJ /F6 7.97 Tf 6.58 0 Td[(+ 2)]TJ /F7 11.955 Tf 11.96 0 Td[(i,1++Z=,0+)]TJ /F5 11.955 Tf 9.74 2.95 Td[(+,0)]TJ /F6 7.97 Tf 6.59 0 Td[(+, (2) 52

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weobtainthefollowingequationsforthereducedsetofvariables 2Z=Fa,0Zd# 2[+)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+)]TJ /F6 7.97 Tf 6.59 0 Td[(+]Z+Fa,0Zd# 2cos#[+)]TJ /F2 11.955 Tf 9.74 1.79 Td[()]TJ /F5 11.955 Tf 11.96 0 Td[()]TJ /F6 7.97 Tf 6.59 0 Td[(+]X4Y=Fa,0Zd# sin2#(+)]TJ /F5 11.955 Tf 9.75 1.79 Td[(+)]TJ /F6 7.97 Tf 6.59 0 Td[(+)+2cos2#++Y4X=Fa,0Zd# cos2#(+)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+)]TJ /F6 7.97 Tf 6.59 0 Td[(+)+2sin2#++X+Fa,0Zd# cos#[+)]TJ /F2 11.955 Tf 9.74 1.79 Td[()]TJ /F5 11.955 Tf 11.96 0 Td[()]TJ /F6 7.97 Tf 6.59 0 Td[(+]Z. (2) TheintegralsinEq. 2 aresolvedexactlytoobtainanalgebraicequationcontainingthespectrumforthreecollectivechiralmodesforarbitraryvaluesofq.ThespectraofthesemodesforFa,0=)]TJ /F5 11.955 Tf 9.3 0 Td[(0.5,areshowninPanel(a)and(b)ofFig. 2-4 .Itisshowninthenextsectionthatthemodeslabelledasx,y,andz,correspondtowavesofmagnetizationsMx,My,andMz,respectively.Thex-modeandy-modesharethesameresonancefrequencyandthespectraforbothrunintotheparticle-holecontinuumatdifferentvaluesofq<=vF,whilethez-modemergeswiththecontinuumatvFq=.Inclusionofhigher-harmonicsoftheLandaufunctionbeyondthes-wave,resultsinextramodeswithhigherresonancefrequencies.However,thehigherthefrequencyofthemode,theheavieritsdampingbyparticle-holeexcitations.Itisinstructivetoanalyzethebehaviorofthecollectivemodesatq!0.ExpandingthesolutionsofEqs. 2 toorderq2,wendforthemodes'dispersions 2i(q)=2(1+Fa,0i)+Diq2,(2) 53

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Figure2-4. a)SpectrumofchiralspinwavesforFa,0=)]TJ /F5 11.955 Tf 9.3 0 Td[(0.5.b)zoomofthesmall-qregionforthex-modeandy-mode.Thedashedcurvesrepresenttheparabolicapproximation.c)Stiffnessesofx,y,z-modesasafunctionofFa,0. 54

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wherex,y=1=2,z=1,andthemodestiffnessesdependonFa,0as Dx=)]TJ /F11 11.955 Tf 11.29 16.86 Td[(2 Fa,0+17 4+13 8Fa,0)]TJ /F5 11.955 Tf 33.47 8.09 Td[((Fa,0)2 16(1+Fa,0=2)v2F (2a) Dy=2 Fa,0+5 4+Fa,0 8+3(Fa,0)2 16(1+Fa,0=2)v2F (2b) Dz=4 Fa,0+13 2+5 2Fa,0v2F. (2c) Atq=0,theseequationsreducetochiralspinresonancesinthes-waveapproximation[ 51 ].Althoughchiralwavesaresomewhatanalogoustospin-wavesinapartiallypolarizedFL[ 70 72 ],theyexhibitanumberofdistinctfeatures.First,chiralspin-waveshaveatleastthreebranches:thisisadirectconsequenceofbrokenSU(2)symmetry:sinceinthein-planeandout-of-planespinsusceptibilitiesarenownotthesame,theirsingularities,correspondingtocollectivemodes,alsooccurindifferentregionsofthe,qplane.Second,theqdependencesofthesex-modesisoftheoppositesigncomparetothatoftheothertwomodes,inawideintervaloftheeecouplingconstant,Fa,0:for)]TJ /F5 11.955 Tf 9.3 0 Td[(0.625
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notationsofRef.[ 51 ]andrepresent^nasasumoftwoterms: ^n=^nSO+^next(2)where^nSO=@"n0^"SOisaperturbationduetotheSOI, ^"SO=(x) 2^epez,(2)whereweallowfortohaveaslow(comparedtoFermiwavelength)positiondependenceintheplaneofthe2Dsystem,n0istheequilibriumFermifunction,and^nextisaperturbationduetoexternalforces.Theexternalpartoftheperturbationcanbeparameterizedas^next=@"n0^upwhere ^up=uip(r,t)^pi,(2)and^pisavectorofrotatedPaulimatrices^p1=)]TJ /F5 11.955 Tf 9.7 0 Td[(^z,^p2=^ep,and^p3=^epez.Matrices^pobeythefollowingalgebra [^p3,^p03]=2i^zsin# (2a) [^p3,^p02]=2i^zcos# (2b) [^p3,^p01]=)]TJ /F5 11.955 Tf 9.3 0 Td[(2i^p2, (2c) where#isdenedbyEq. 2 .AchangeintheQPenergyisgivenby ^"=^"SO)]TJ /F3 11.955 Tf 13.15 8.08 Td[(F 2Tr0Zd0 2^f(p,p0)^u0(2)where^f(p,p0)istheLandaufunction,=2pF,F=m=isthedensityofstates,andprimereferstothequantumnumbersofanelectronwithmomentump0.NoticethattheTr0R^f(p,p0)^nSOdp0contributionto^"isincludedviarenormalizationoftheSOcoupling!==(1+Fa,1=2),whereFa,1istherstharmonicofthespinpartoftheLandaufunction[ 38 51 ].Thecomponentsofmagnetizationareexpressedvia^up, 56

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projectedontotheFS,as Mi=gB 4FZd 2Tr(^i^upF).(2)Expandingupioverangularharmonicsas upi=Xnu,nicosnp+u,nisinnp,(2)weobtainforthedimensionlesscomponentsofmagnetizationwhere Mx=1 2(u,12+u,13)My=1 2(u,12)]TJ /F7 11.955 Tf 11.95 0 Td[(u,13)Mz=u,01 (2) withMx,y=(gBF=2)Mx,yandMz=)]TJ /F5 11.955 Tf 9.29 0 Td[((gBF=2)Mz.ToleadingorderinSO,thecollisionintegralduetoscatteringatshort-rangeimpuritiescanbewrittenas @^n @tcoll=)]TJ /F5 11.955 Tf 10.59 8.09 Td[(^n)]TJ /F5 11.955 Tf 12.06 2.66 Td[(^n ,(2)where^ndenotestheangularaverage,istheimpuritymean-freetime.Afterallthesesteps,inthes-waveapproximation,weobtainasystemofcoupledBoltzmannequations (^)]TJ /F14 11.955 Tf 11.95 0 Td[(v^q)iup1)]TJ /F5 11.955 Tf 11.95 0 Td[(up2=Fa,0(cospMx+sinpMy)+Fa,0v^q+i iMz (2) (^)]TJ /F14 11.955 Tf 11.95 0 Td[(v^q)up2)]TJ /F5 11.955 Tf 11.95 0 Td[(iup1=Fa,0iMz+Fa,0v^q+i (cospMx+sinpMy) (2) (^)]TJ /F14 11.955 Tf 11.95 0 Td[(v^q)up3=Fa,0v^q+i (sinpMx)]TJ /F5 11.955 Tf 11.96 0 Td[(cospMy) (2) where^i@t+i=,and^q)]TJ /F7 11.955 Tf 22.45 0 Td[(irr.Theright-handsidesofEqs. 2 2 containonlythelowestangularbasisfunctions,whichallowsonetoderiveclosedequationsfor 57

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theharmonicsofupior,equivalently,forthecomponentsofmagnetization: 2iMz=Fa,0Zdp 2^P+)]TJ /F5 11.955 Tf 9.07 1.63 Td[((iMz+cospMx+sinpMy)+^P)]TJ /F6 7.97 Tf 6.59 0 Td[(+(iMz)]TJ /F5 11.955 Tf 11.95 0 Td[(cospMx)]TJ /F5 11.955 Tf 11.95 0 Td[(sinpMy), (2a) 4Mx=Fa,0Zdp cosp)]TJ /F5 11.955 Tf 7.99 -7.16 Td[(^P+)]TJ /F2 11.955 Tf 9.74 1.79 Td[()]TJ /F5 11.955 Tf 14.47 2.53 Td[(^P)]TJ /F6 7.97 Tf 6.58 0 Td[(+iMz+Fa,0Zdp cos2p)]TJ /F5 11.955 Tf 7.99 -7.16 Td[(^P+)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+^P)]TJ /F6 7.97 Tf 6.58 0 Td[(++2sin2p^P++Mx+Fa,0Zdp cospsinp)]TJ /F5 11.955 Tf 7.99 -7.16 Td[(^P+)]TJ /F5 11.955 Tf 9.75 1.79 Td[(+^P)]TJ /F6 7.97 Tf 6.58 0 Td[(+)]TJ /F5 11.955 Tf 11.96 0 Td[(2^P++My, (2b) 4My=Fa,0Zdp sinp)]TJ /F5 11.955 Tf 7.99 -7.16 Td[(^P+)]TJ /F2 11.955 Tf 9.75 1.79 Td[()]TJ /F5 11.955 Tf 14.46 2.52 Td[(^P)]TJ /F6 7.97 Tf 6.58 0 Td[(+iMz+Fa,0Zdp sinpcosp)]TJ /F5 11.955 Tf 7.99 -7.16 Td[(^P+)]TJ /F5 11.955 Tf 9.75 1.79 Td[(+^P)]TJ /F6 7.97 Tf 6.58 0 Td[(+)]TJ /F5 11.955 Tf 11.96 0 Td[(2^P++Mx+Fa,0Zdp sin2p)]TJ /F5 11.955 Tf 7.98 -7.16 Td[(^P+)]TJ /F5 11.955 Tf 9.74 1.79 Td[(+^P)]TJ /F6 7.97 Tf 6.59 0 Td[(++2cos2p^P++My (2c) where ^P+)]TJ /F5 11.955 Tf 10.41 1.79 Td[(=(^)]TJ /F5 11.955 Tf 11.96 0 Td[()]TJ /F14 11.955 Tf 11.96 0 Td[(v^q))]TJ /F6 7.97 Tf 6.59 0 Td[(1(+v^q+i Fa,0).(2)Theoperators^P)]TJ /F6 7.97 Tf 6.59 0 Td[(+and^P++areobtainedfrom^P+)]TJ /F1 11.955 Tf 10.4 1.8 Td[(bysubstituting!)]TJ /F5 11.955 Tf 25.1 0 Td[(and=0,respectively.Notethat(x)and^qdonotcommute;therefore,theorderofoperatorsmustbekeptintact.KeepingMitotherightof^Pss0emphasizesoperationsinspaceandtime.Moreover,wechoose^qtobealongthex-axis.SeveralintegralsinEqs. 2a 2b ,and 2c vanishduetothischoice,andasaresultMyisdecoupledfromtheMx-Mzsystem.Atq=0,Eqs. 2a 2b ,and 2c describechiralresonancesatfrequenciesx,yandz,whicharethestartingpointsofthedispersionrelationsforthecollectivemodes.Forsmallq,themodefrequenciesarestillclosetoxandz.Toobtaindispersionrelationinthisregion,weformallyexpandEqs. 2a 2b 58

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and 2c toorder^q2x.Foreigenvalues,itmeansthatweareworkingintheregimevqmaxf)]TJ /F5 11.955 Tf 11.96 0 Td[(,1=g,forinstance ^P+)]TJ /F5 11.955 Tf 17.05 1.79 Td[(= )]TJ /F5 11.955 Tf 11.96 0 Td[(+ ()]TJ /F5 11.955 Tf 11.96 0 Td[()2v^q+ ()]TJ /F5 11.955 Tf 11.95 0 Td[()3(v^q)2+ ()]TJ /F5 11.955 Tf 11.96 0 Td[()4v(^q)v^q, (2) where^qin(^q)actsonlyon(x).Using)]TJ /F6 7.97 Tf 6.59 0 Td[(1(x)vF^qx!^qxand^=(x)!^,wendthat 2iMz=Fa,0 22 2)]TJ /F5 11.955 Tf 11.96 0 Td[(1+1 ()]TJ /F5 11.955 Tf 11.96 0 Td[(1)2+1 (+1)2^qxMx+i 21 ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)2)]TJ /F5 11.955 Tf 32.33 8.09 Td[(1 (+1)2^qxMx+2"Fa,0+i 2)]TJ /F5 11.955 Tf 11.96 0 Td[(1#iMz+Fa,0+i 21 ()]TJ /F5 11.955 Tf 11.96 0 Td[(1)3+1 (+1)3^q2xiMz+Fa,0 21 ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)4)]TJ /F5 11.955 Tf 32.33 8.09 Td[(1 (+1)4(^qx)^qxiMz (2) 4Mx=Fa,02 2)]TJ /F5 11.955 Tf 11.96 0 Td[(1+1 ()]TJ /F5 11.955 Tf 11.96 0 Td[(1)2+1 (+1)2^qxiMz+i 1 ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)2)]TJ /F5 11.955 Tf 32.33 8.09 Td[(1 (+1)2^qxiMz+2"i +Fa,0+i 2)]TJ /F5 11.955 Tf 11.96 0 Td[(1#Mx+3 4(Fa,0+i )1 ()]TJ /F5 11.955 Tf 11.96 0 Td[(1)3+1 (+1)3^q2xMx+"Fa,0+i 22#^q2xMx+3Fa,0 41 ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)4)]TJ /F5 11.955 Tf 32.34 8.09 Td[(1 (+1)4(^qx)^qxMx (2) 4My=2"i +Fa,0+i 2)]TJ /F5 11.955 Tf 11.96 0 Td[(1#My+Fa,0+i 41 ()]TJ /F5 11.955 Tf 11.96 0 Td[(1)3+1 (+1)3+6 3^q2xMy+Fa,0 41 ()]TJ /F5 11.955 Tf 11.95 0 Td[(1)4)]TJ /F5 11.955 Tf 32.33 8.09 Td[(1 (+1)4(^qx)^qxMy. (2) Toobtainmacroscopicequationsofmotionintheballisticlimit,onetakesthelimitof!1andsolvethesystemof 2 2 ,and 2 toleadingorderin^qx.The 59

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solutionsdescribethethreemodesdiscussedinthelastsection.Inthez-mode, )]TJ /F3 11.955 Tf 11.95 0 Td[(@t2Mz=2(1+Fa,0)Mz)]TJ /F7 11.955 Tf 11.95 0 Td[(Dz@x2Mz)]TJ /F7 11.955 Tf 11.96 0 Td[(Cz0 @xMz, (2a) Mx=2+2Fa,0 Fa,0vF@xMz, (2b) where0=d=dx,themodestiffnessDicanbereadfromEqs. 2a 2c ,and Cz=2 (Fa,0)2+1 Fa,01 Fa,0+1v2F.(2)Thez-modeisawaveoftheout-of-planemagnetizationpropagatingin-plane.However,atniteq,asmallMx(/vFq Mz)magnetizationcomponentisdevelopedalongthedirectionofpropagation.Thex-modeisdescribedbythefollowingequationsofmotion )]TJ /F3 11.955 Tf 11.95 0 Td[(@t2Mx=21+Fa,0 2Mx)]TJ /F7 11.955 Tf 11.96 0 Td[(Dx@2xMx)]TJ /F7 11.955 Tf 11.96 0 Td[(Cx0 @xMx, (2a) Mz=4+2Fa,0 Fa,0vF@xMx, (2b) where Cx=122 (Fa,0)2+1 2Fa,01 Fa,0+1 2v2F.(2)ThischiralmodedescribesawaveoftheMxmagnetizationpropagatinglongitudinallyinthexdirection.However,atniteq,asmallMz(/vFq Mx)magnetizationcomponentisproduced.Inthey-mode, )]TJ /F3 11.955 Tf 11.96 0 Td[(@t2My=21+Fa,0 2My)]TJ /F7 11.955 Tf 11.95 0 Td[(Dy@2xMy)]TJ /F7 11.955 Tf 11.95 0 Td[(Cy0 @xMy, (2) whereCy=Cx=3.Thischiralmodeisawaveofin-planemagnetizationMypropagatingtransverselyinthexdirection. 60

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Themaindampingeffectiscontainedintheq=0limitofEqs. 2 and 2 ,bywhichonearrivesatEqs. 2a and 2b .Inthepresenceofdamping,themodesarebroadenedalmostindependentofthewavelength.Thereforetheeffectofdampingismainlycontainedintheq=0formoftheequationsofmotion )]TJ /F3 11.955 Tf 11.95 0 Td[(@t@t+1 2Mx,y=21+Fa,0 2@t+1+Fa,0 2Mx,y. (2a) )]TJ /F3 11.955 Tf 9.3 0 Td[(@t@t+1 Mz=2(1+Fa,0)Mz. (2b) TheseequationsdescribeDyakonov-Perelspinrelaxation[ 73 ]renormalizedbytheeeinteraction.Themodesarewellresolvedintheballisticlimit,1.Anothersourceofdampingforchiralspin-wavesisduetoelectron-electronscattering,characterizedby 1 eeAvFq 2[2+2T2],(2)whereAisthepre-factorintheimaginarypartoftheelectronself-energy,whichinaFLscalesasIm(!,T)=A(!2+2T2).InaGaAsheterostructurewithnumberdensityn1011cm)]TJ /F6 7.97 Tf 6.59 0 Td[(2,usingexperimentalresultsinRef.Eisenstein1995,weestimatethatA=0.1meV)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Insuchsystem,=0.1meV.[ 74 ]Therefore,giventhatinthelongwavelengthregimevFq,thedampingduetoe-ecollisionsarenegligiblecomparedtothedampingduetodisorder.ThetwodampingsourcesarecomparableonlyathightemperaturesT=vFqp A. 2.6.3StandingchiralwavesForstanding-wavesolutions,Miexp(iit),Eqs. 2a 2a ,and 2 aretransformedintotheSchroedingerequationsformassiveparticles )]TJ /F5 11.955 Tf 16.66 8.08 Td[(1 2mi@2x+Vi(x)Mi=EiMi,(2) 61

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wherei=fx,y,zg,theeffectivemassesarerelatedtothestiffnessesinEqs. 2a 2b ,and 2c viami=1=2Di,Ei=2i,andVi(x)=2(x)(1+Fa,0i)arethepotentialenergies,whichwenowallowtovaryslowly(comparedtotheelectronwavelength)inthe2DEGplane.Bymeansofthelateralproleof,thepotentialViconnesthespin-chiralmodes.Theseconnedmodes,i.e.standingchiralwaves,aredescribedbytheboundstatesofEq. 2 .TheextratermduetothegradientofisignoredinthepotentialVi.ThesetermsareconsideredinSec. 2.6.4 ,whereweshowthattheycausenoqualitativechangeintheenergylevelsfortheboundstateswhileaddingaslowlyvaryingenvelopetotheMiwavefunction,i.e.Mi!MiebVi(x)=2,wherebisaconstantproportionaltotherelativestrengthofthegradientterm-seeSec. 2.6.4 .However,thesechangesdonotaffectthequalitativeargumentspresentedinthissection.Theeffectivemassesofthez-modeandthey-modearenegativeforanyFa,0withintheintervalfrom)]TJ /F5 11.955 Tf 9.3 0 Td[(1to0.Therefore,thez-andy-modes,asshowninthebottompartofFig 2-5 b,canbeconnedbyapotentialbarrierin.Theeffectivemassofthex-modeispositiveforFc
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Figure2-5. a)Sketchofthesuggestedexperimentalsetup.TopgatemodulatestheSOsplitting.b)Top:Aminimuminconnesthex-modewithapositiveeffectivemass.Bottom:Amaximuminconnesthez-mode.Thecorrespondingmicrowaveabsorptionspectraareshownschematicallyinarbitraryunits. hence,reversingthepolarity,wouldeitherstronglysuppressorenhancetheabsorptionspectrum.However,inallcasesthetwochiralspinresonancesshouldbeobservedasstrongpeaksastheyexistevenintheabsenceoftheinteraction,andthespin-chiralwaves. 2.6.4Parabolicconnementofchiralspin-wavesInSec. 2.6.3 ,theequationsofmotionforstandingwavesolutionsofchiral-spincollectivemodesaretransformedintoaSchrdinger-likeequation-seeEq. 2 .However,thepotentialtermsinvolvingthegradientofthe(x)areignoredinthatsection.Herewerestorethegradienttermsandassesstheireffect.UsingMi(x,t)=Mi(x)exp(iit),Eqs. 2a 2a ,and 2 canbewrittenas 2iMi=)]TJ /F5 11.955 Tf 16.67 8.09 Td[(1 2mi@2x+2(x)(1+Fa,0i))]TJ /F7 11.955 Tf 11.95 0 Td[(Ci0 @xMi,(2)whichcaninturnbewrittenas "iMi=)]TJ /F5 11.955 Tf 10.5 8.08 Td[(1 2@2x+1 2U(x)+bU0(x)@xMi,(2) 63

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where"i=mi[2i)]TJ /F5 11.955 Tf 11.97 0 Td[(20(1+Fa,0i)],U=2mi[2)]TJ /F5 11.955 Tf 11.97 0 Td[(20](1+Fa,0i),U0(x)=dU(x)=dx,andb=Ci=22(1+Fa,0i)istakentobeaconstant.OnecaneliminatethegradienttermbylettingMi(x)!Mi(x)ebU(x)=2andarriveat "iMi=)]TJ /F5 11.955 Tf 10.5 8.08 Td[(1 2@2x+1 2U(x)+b2 4U02(x))]TJ /F7 11.955 Tf 13.15 8.08 Td[(b 2U00(x)Mi.(2)ThismeansthatthegradienttermsforceaGaussianenvelopetothestandingchiralspin-wavesandalsoslightlymodifytheirenergy.Forthecaseofaparabolicconnement,i.e.U(x)=hx2, "i+bh 2Mi=)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 2@2xMi+1 2)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(1+b2h2x2Mi,(2)whichdescribesaharmonicoscillatorwitharenormalizedfrequencyandshiftedenergylevels: "i,n=p 1+b2h2n+1 2)]TJ /F7 11.955 Tf 13.15 8.08 Td[(bh 2.(2) 2.6.5ConditionsforobservationTodistinguishbetweenthechiralresonancesandthemodesdescribedhere,oneneedstoobserveseveralboundstates,whichrequirestheproleof(x)tosatisfycertainrequirements.First,arealisticpotentialhasanitedepth,Wwhosevaluedeterminesthenumberofpossibleboundstates.SupposeVi=20(1+Fa,0i))]TJ /F7 11.955 Tf -423.43 -23.91 Td[(W=cosh2(x x0),where0=(x=0).Atxx0,thispotentialisidenticaltoaparabolicpotential Vi=20(1+Fa,0i)+1 2mi!2Hx2,(2)where!H2p W=mix20.Tohavemorethanoneboundstate,W>!H=2.[ 75 ]Therefore,theconditiontohavemorethanoneboundstateamountstomix20W>1.Furthermore,fortheboundstatetoexist,itshouldnotbedampedbyparticle-holeexcitations.Tondspecicconditions,letusconsideraparabolicconnementasinEq. 2 TheboundstatesforthiscasearefoundinSec. 2.6.4 .Thecharacteristicsizeof 64

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theseboundstatesarea1=p mi~!H,where~!H=!Hp 1+(mi!Hb)2therenormalizedfrequencyduetothegradientterms.ForFa,0=)]TJ /F5 11.955 Tf 9.29 0 Td[(0.5,qx,y0.2=SO(cf.Fig. 2-4 ),whereSO1=2mjj,andqz=1=SOindependentofFa,0.[ 76 ]InaGaAsheterostructurewith=5meVA,[ 74 ]SO1m.Similarly,theconditiontoobserveatleasttwoboundstatesisthatashouldexceedp 3qi.Thisconditionisobtainedusingtheclassicalsizeofthewavefunctionforn=1levelofharmonicoscillator.UsingtheabovevaluesforSOandFa,0,a&2m(a&9m)forobservationofthez-mode's(x-andy-modes')boundstates.Thisconditionalsoimpliesanupperlimiton=p !H,i.e.thespacingbetweentheabsorptionpeaks: i06qiSO p 3s Di (1+Fa,0i)v2F,(2)wherei0aretheresonancefrequenciesatq=0.Anothernecessaryconditionforobservationoftheboundstatesisthattheirbroadeningwidth,whichisoforder1=intheballisticregimeshouldnotexceedtheirlevel-spacing.Thereforetheconditiontohavewell-resolvedboundstatesamountsto1=.ForjFa0j1,thisconditionisthesameastheconditionfortheballisticlimit,i.e.,1.Ignoringthedependenceofonthenumberdensityn,inaGaAsheterostructurewendthat=3.510)]TJ /F6 7.97 Tf 6.58 0 Td[(6p n[1011cm)]TJ /F6 7.97 Tf 6.59 0 Td[(2][cm2=Vs],whereisthecarriermobility.Tosatisfytheconditionfortheballisticregimemustexceed106cm2=Vs. 2.6.6Excitationofchiralspin-wavesTheobviouswaytoexcitethechiralspinwavesofMiisbymeansofamagneticeldHi,oscillatingnearthecorrespondingresonancefrequency.Oneneedstosimplyaddtheterm)]TJ /F7 11.955 Tf 9.3 0 Td[(gBHiitotherighthandsideofEq. 2 andproceedsimilarlykeepingHionlytolinearorder,whichresultsin ^LiMi=g2 42BF2iHi,(2) 65

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where^Li=)]TJ /F3 11.955 Tf 9.3 0 Td[(@t2)]TJ /F5 11.955 Tf 10.18 0 Td[(2(1+iFa,0)+Di@2x.However,theSOinteractionallowsforacouplingofspinstoanin-planeelectriceld,E,whichareexperimentallyeasiertoimplementandmanipulate.Inordertoincludetheexternalelectriceldintotheequationofmotion,thecanonicalmomentump!p)]TJ /F7 11.955 Tf 12.36 0 Td[(eA=c,whereA=icE=!,and!=)]TJ /F7 11.955 Tf 9.3 0 Td[(i@t.Toincludetheeffectoftheexternaleldintheequationofmotionforchiralspin-waves,itsufcestoonlyaddthetensorpartoftheeld-dependentHamiltonian ^"E=(^ez)eA c,(2)totherighthandsideofEq. 2 .FortheMxmodeonlythelinearterminEcanbekeptwhichappearsinonlythefollowingcommutation ^"SO,(^ez)eA c=e !(^pE)ezz,(2)asapartof[^n,^"]intheBoltzmannequation.Consequentlyonearrivesatthefollowingequation, ^LxMx=)]TJ /F7 11.955 Tf 10.49 8.08 Td[(gB 4F2eEy x0, (2a) ^LxMy=+gB 4F2eEx x0, (2b) where0iaretheresonancefrequenciesatq=0.FortheMzmode,however,thelinearorderterminEisabsentandthehigherorderscanbekeptthroughthefollowingcommutation ^u,(^ez)eA c,(2)as^uisalreadyproportionaltotheexternaleld.Consequently,onearriveat ^LzMz=41)]TJ /F7 11.955 Tf 13.15 8.09 Td[(Fa02 2z0e z0jE^zj2Mz. (2) Equation( 2 )describesaparametricresonanceinMzexcitedbytheelectriceldwithfrequencyz0.TheinitialamplitudeofMzcanbeprovidedbyapulsein 66

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Bz.Therefore,whilethex-modecoupleslinearlytotheelectriceld,thez-modeisgeneratedtosecondorderinE.ItisworthnotingthatalloftheresultspresentedaboveremainthesameiftheRashbaSOinteractionisreplacedbytheDresselhausone.IftheRashbaandDresselhausinteractionsarepresentsimultaneously,spin-chiralmodesbecomenon-sinusoidal.Hence,itisbettertoperformtheexperimentonasymmetricquantumwellwhichhasonlytheDresselhausbutnoRashbainteraction. 67

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CHAPTER3CONCLUSIONSA2DFLsystemofelectronsinthepresenceofRashbaSOinteractionisstudied.ThepresenceofSObreakstheSU(2)symmetryofthesystemandonthebasisoftheremainingsymmetriesasuitableLandaufunctionforthissystemisconstructedinEq. 2 .InSec. 2.5.2 ,thisparticularformoftheLFwasalsoobtainedbyasecondorderperturbationtheoryanalysis.ThekeyresultofthisdissertationisthatwhilethechargesectorofachiralFLcanbefullydescribedbytheLandaufunctionprojectedonthetwospin-splitFSs,anyquantitypertainingtothespinsectormustinvolvetheLandaufunctionwithmomentainbetweenthetwoFSs.Thisfeatureismostexplicitlydemonstratedforthecaseofthestatic,out-of-planespinsusceptibility.Therefore,thereisnoconventionalFLtheory,i.e.,atheoryoperatingsolelywithfreequasiparticles,forthespinsectorofachiralFL.Thisdoesnotmeanthatwearedealingherewithanon-Fermiliquid,becausechiralquasiparticlesarestillwell-denedneartheirrespectiveFSs.However,thespinsectorofachiralFLdoesnotallowforaFL-typedescription.Inotherwords,wearedealingwithaspecialclassofFLs(non-LandauFLs),whichareFLswithoutafull-edgedFLtheory.NotonlychiralbutalsoanyFLwithbrokenSU(2)symmetry,e.g.,a3DFLinthepresenceofnitemagneticeld,belongstothisclass.[ 57 ]Thepropertiesofthissysteminthechargesectoraresimilartoanytwobandsystem.OneinterestingeffectoftheinteractionsinasystemwithpureRashbaorDresselhausSOinteraction,istobreakthedegeneracyofvelocitiesoffermionswithdifferentchiralities.AsthephenomenologicalformulafortheeffectivemassesEq. 2 suggests,theeffectivemassesandvelocitiesofRahsbaFermionsaresplitduetotheeffectofinteractions.WeobtaintheleadingsplittingforthecaseofColoumbinteractionsinEq. 2 ,aswellasinAppendix A .Inaddition,inamodelwithshort-rangeinteractionsweevaluatetheleadingnon-analyticcorrectiontotheLandau 68

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parametersandshowthattheyareoftheorderU22,whereUistheamplitudeoftheshort-rangeinteraction.Inthespinsector,usinglinearresponseofaSU2SFLtoanSOperturbation,anewtypeofcollectivemodeispredicted:spin-chiralwavesinaFLwithRashbaSOcoupling[ 77 ].Alreadyinthefreecase,precessionoftheelectronspinaroundtheRashbaSOeldleadstoabsorptionofelectromagneticradiationatthefrequencygivenbytheRashbasplittingofchiralsubbands.Thisisaresonanceinzeromagneticeld[ 15 78 ].Theeeinteractionaddsseveralqualitativelydifferentelementstothisstory.First,itbreaksspin-rotationalinvariance;hencethefrequenciesofnormalmodeswithin-andout-of-planecomponentsofthemagnetizationaredifferent.Thisisaninteraction-inducedsplittingofthespin-chiralresonance[ 51 ].Second,theresonancefrequenciesacquiredispersion:thesearespin-chiralwavesthatexistonlyinthepresenceofbotheeandSOinteractions[ 77 ].Thespectrumandequationsofmotionforchiralspinwavesareobtained.Inaddition,weproposeanexperimenttoobservethesemodesvialateralmodulationsoftheSOcoupling.AlltheresultsofpresentedherealsoholdforasystemwithDresselhausSOinteraction.However,ifbothRashbaandDresselhausarepresent,theSO(2)symmetryoftheFSsisbrokenaswell,manifestingitselfincomplicationssuchasthepresenceofmanymoreLandauparametersinphenomenologicalresults,non-sinusoidalspin-waves,etc.Ourresultscanalsobesimplygeneralizedtothecaseofholesystems.TheHamiltonianforfreeRashbaheavy-holescanbecastintothefollowingfrom ^Hholes=p2 2m1+i 2)]TJ /F5 11.955 Tf 6.66 -9.68 Td[(^+p3)]TJ /F2 11.955 Tf 9.74 2.96 Td[()]TJ /F5 11.955 Tf 13.14 0 Td[(^)]TJ /F14 11.955 Tf 7.08 1.8 Td[(p3+,(3) 69

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wherep=pxipy,andsimilarlyfor^.Equation 2 doesnotholdfora2DFLsystemofheavy-holes.Infactthefollowingform ^fholes=fs110+fak(^x^0x+^y^0y)+fa?^z^0z+1 2gphh1^0+(p0)]TJ /F6 7.97 Tf 8.09 9.91 Td[(3)]TJ /F14 11.955 Tf 11.95 0 Td[(p)]TJ /F6 7.97 Tf 8.09 6.73 Td[(3))]TJ /F5 11.955 Tf 13.14 0 Td[(^0)]TJ /F5 11.955 Tf 7.08 2.96 Td[((p0+3)]TJ /F14 11.955 Tf 11.95 0 Td[(p+3))]TJ /F5 11.955 Tf 20.45 0 Td[(10^+(p0)]TJ /F6 7.97 Tf 8.1 9.91 Td[(3)]TJ /F14 11.955 Tf 11.96 0 Td[(p)]TJ /F6 7.97 Tf 8.09 6.73 Td[(3))]TJ /F5 11.955 Tf 13.14 0 Td[(^)]TJ /F5 11.955 Tf 7.08 1.79 Td[((p0+3)]TJ /F14 11.955 Tf 11.96 0 Td[(p+3)i+1 2gpph1^0+(p0)]TJ /F6 7.97 Tf 8.09 9.9 Td[(3+p)]TJ /F6 7.97 Tf 8.09 6.73 Td[(3))]TJ /F5 11.955 Tf 13.14 0 Td[(^0)]TJ /F5 11.955 Tf 7.08 2.95 Td[((p0+3+p+3)+10^+(p0)]TJ /F6 7.97 Tf 8.1 9.91 Td[(3+p)]TJ /F6 7.97 Tf 8.09 6.73 Td[(3))]TJ /F5 11.955 Tf 13.14 0 Td[(^)]TJ /F5 11.955 Tf 7.08 1.8 Td[((p0+3+p+3)i+h(1))]TJ /F5 11.955 Tf 6.66 -9.68 Td[(^+p3)]TJ /F2 11.955 Tf 9.74 2.95 Td[()]TJ /F5 11.955 Tf 13.14 0 Td[(^)]TJ /F14 11.955 Tf 7.09 1.79 Td[(p3+^0+p03)]TJ /F2 11.955 Tf 9.74 2.95 Td[()]TJ /F5 11.955 Tf 13.14 0 Td[(^0)]TJ /F14 11.955 Tf 7.09 2.95 Td[(p03++h(2))]TJ /F5 11.955 Tf 6.66 -9.69 Td[(^0+p3)]TJ /F2 11.955 Tf 9.74 2.95 Td[()]TJ /F5 11.955 Tf 13.14 0 Td[(^0)]TJ /F14 11.955 Tf 7.09 2.95 Td[(p3+^+p03)]TJ /F2 11.955 Tf 9.74 2.95 Td[()]TJ /F5 11.955 Tf 13.14 0 Td[(^)]TJ /F14 11.955 Tf 7.09 1.79 Td[(p03++1 2h)]TJ /F5 11.955 Tf 11.64 -9.68 Td[(^+p3)]TJ /F2 11.955 Tf 9.75 2.96 Td[()]TJ /F5 11.955 Tf 13.13 0 Td[(^)]TJ /F14 11.955 Tf 7.09 1.8 Td[(p3+)]TJ /F5 11.955 Tf 14.14 -9.68 Td[(^0+p3)]TJ /F2 11.955 Tf 9.74 2.96 Td[()]TJ /F5 11.955 Tf 13.14 0 Td[(^0)]TJ /F14 11.955 Tf 7.08 2.96 Td[(p3++^+p03)]TJ /F2 11.955 Tf 9.74 2.95 Td[()]TJ /F5 11.955 Tf 13.14 0 Td[(^)]TJ /F14 11.955 Tf 7.08 1.79 Td[(p03+^0+p03)]TJ /F2 11.955 Tf 9.74 2.95 Td[()]TJ /F5 11.955 Tf 13.14 0 Td[(^0)]TJ /F14 11.955 Tf 7.08 2.95 Td[(p03+i, (3) canserveastheappropriateLFforaFLsystemofholes.Themajordifferencefromthecaseofelectronliquidsisthecouplingtomagneticeldssuchasinlinearspinsusceptibilities,wherethelinearZeemantermforholesisoftheform)]TJ /F5 11.955 Tf 6.67 -9.68 Td[(^+p2)]TJ /F14 11.955 Tf 7.08 2.96 Td[(B)]TJ /F5 11.955 Tf 9.74 1.8 Td[(+^+p2+B+.However,thestructureofthetheoryremainsqualitativelythesame. 70

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APPENDIXAMASSRENORMALIZATIONVIATHESELF-ENERGYInSec. 2.3 ,weanalyzedmassrenormalizationofRashbafermionstologarithmicaccuracyintheparameterpTF=pwhichcontrolstheRandomPhaseApproximationfortheCoulombinteraction,andshowedthatdegeneracyofthesubbands'Fermivelocitiesisliftedbytheeeinteraction,atleastwhenSOcouplingissostrongthatp+p)]TJ /F1 11.955 Tf -427.21 -22.11 Td[((butstillbothsubbandsareoccupied).InthisAppendix,wederivetheresultformassrenormalizationstilltoleadingorderintheeeinteractionbutforanarbitrarilystrongSOinteraction.Forthetimebeing,weassumethattheinteractionisdescribedbyanon-retardedandspin-independent,butotherwisearbitrarypotentialUq.Toobtaintheresultbeyondthelogarithmicaccuracy,oneneedstokeepthematrixelementsinEq.( 2 )andrelaxtheassumptionofsmallmomentumtransfers.Theonlysmallparameterwillnowbethequasiparticleenergy,"p,fs=p,fs)]TJ /F3 11.955 Tf 12.43 0 Td[(.Fromthispointon,wesuppressthesuperscriptfsincealldispersionsconsideredbelowarefornon-interactingelectrons,i.e,"p,fs!"ps.Theleadingorderrenormalizationislogarithmic,thereforethepotentialcanbetotakenasstaticsincethedynamicpartdoesnotproducelogarithmicintegrals.SubtractingfromEq.( 2 )theself-energy,evaluatedontheFS,weobtainfortheremainder s=)]TJ /F11 11.955 Tf 11.29 11.36 Td[(Xs0Zd2q (2)2Zd 2Uq 2[(1+ss0cosp+q)gs0(p+q,!+))]TJ /F5 11.955 Tf 19.26 0 Td[((1+ss0cosps+q)gs0(ps+q,)]. (A) Becauseofthein-planerotationalsymmetry,itisconvenienttomeasureallanglesfromthedirectionofpwhich,bydenition,coincideswiththedirectionofps.Usinganexactresultsp+qs0=jp+qj2 2m+s0jp+qj=p2 2m+pq mcos+q2 2m+s0jp+qj, (A) 71

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whereistheanglebetweenpandq,andexpandingjp+qjtolinearorderin"psasjp+qj=ps+"ps v0ep+q=jps+qj+"ps vcos(ps+q), (A)wherev0isgivenbyEq.( 2 ),weget"p+qs0="ps1+q mv0cos)]TJ /F7 11.955 Tf 13.15 8.08 Td[(s v0+s0cosps+q+psq mcos+q2 2m+s0jps+qj)]TJ /F7 11.955 Tf 17.93 0 Td[(sps, (A)andcosp+q=cosps+q+m"ps psjps+qjsin2ps+q. (A)Onecandecomposesintotwopartsas=1s+2swith 1s=)]TJ /F11 11.955 Tf 11.29 11.35 Td[(Xs0Zd2q (2)2Zd 2Uq 2(1+ss0cosps+q)[gs0(p+q,!+))]TJ /F7 11.955 Tf 11.95 0 Td[(gs0(ps+q,)], (A) 2s=)]TJ /F7 11.955 Tf 10.49 8.09 Td[(m"ps psXs0ss0Zd2q (2)2Zd 2Uq 2sin2ps+q jps+qjgs0(ps+q,!+), (A) wheretherstpartcomesfromonlytheimmediatevicinityoftheFS,whilethesecondonecomesfromtheentireband.IntegratingoverinEq.( A ),weobtainthedifferenceoftwoFermifunctionsofthes0subband.ExpandingtherstFermifunctionwiththehelpofEq.( A ),weget s1="psXs0Zd2q (2)2U(q) 2(1+ss0cosps+q)1+q mv0cos)]TJ /F7 11.955 Tf 13.15 8.08 Td[(s v+s0cosps+qpsq mcos+q2 2m+s0jps+qj)]TJ /F7 11.955 Tf 17.93 0 Td[(sps. (A) 72

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The-functionin( A )enforcesthemass-shellconditionjps+qj=ps0,fromwhichwendcos=)]TJ /F7 11.955 Tf 15.61 8.09 Td[(q 2ps+(s)]TJ /F7 11.955 Tf 11.96 0 Td[(s0)m q1+sm ps, (A)cosps+q=1)]TJ /F7 11.955 Tf 22.51 8.09 Td[(q2 2psps0+js)]TJ /F7 11.955 Tf 11.96 0 Td[(s0j(m)2 psps0, (A)whereweusedthatcosp+q=(p+qcos)=jp+qjforarbitrarypandq.Integratingover,wearriveat1s="ps 22v0"Z2ps0dqUqs 1)]TJ /F7 11.955 Tf 16.19 8.09 Td[(q2 4p2s1)]TJ /F7 11.955 Tf 25.84 8.09 Td[(q2 2psmv0)]TJ /F7 11.955 Tf 13.15 8.09 Td[(s vq2 2p2s+Z2mv02mdqUqq2 4p2F)]TJ /F11 11.955 Tf 11.96 13.27 Td[(m pF21)]TJ /F4 7.97 Tf 22.31 5.03 Td[(q2 2psmv0+s v0q2 2p2F+2m2 v0ps)]TJ /F7 11.955 Tf 11.95 0 Td[(s2m23 v0p2F h1)]TJ /F4 7.97 Tf 13.15 4.7 Td[(s v01+ps p)]TJ /F21 5.978 Tf 5.76 0 Td[(sir 1)]TJ /F11 11.955 Tf 11.95 13.27 Td[(hq 2ps)]TJ /F7 11.955 Tf 11.95 0 Td[(s2m q1+sm psi23775, (A)wheretherst(second)termisaresultofintra-band(inter-band)transitions.Equation( A )isexactin.IfUqisreplacedbythescreenedCoulombpotentialU(q)=2e2=(q+pTF),wherepTF=p 2rspFistheThomas-Fermiscreeningmomentum,therstterminEq.( A )produces,inthesmall-rslimit,therslnrsresultofEq.( 2 ): 1s="ps 22v0Z2ps0dqUTFq=ps vFe2ln2ps pTF+OpTF ps.(A)Notethat,incontrasttoEq.( 2 ),wherepsoccursasanuppercutoffalogarithmicallydivergentintegral,theupperlimitoftheintegralinEq.( A )isdeneduniquely.Equation( A )alsoallowsonetostudythedependenceoftheself-energyonatsmall,withoutassumingthatthisdependenceisanalytic.Torstorderin,theintegrandcanbeexpandedas1s=)]TJ /F3 11.955 Tf 17.37 8.09 Td[("ps 2vFs vFZ2pF0dqUqq2 2p2Fs 1)]TJ /F7 11.955 Tf 16.84 8.09 Td[(q2 4p2F. (A) 73

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wherepFandvFaretheFermimomentumandFermivelocityat=0,correspondingly.Ontheotherhand,integrationoverinEq.( A )gives2s=)]TJ /F7 11.955 Tf 10.49 8.09 Td[(m"ps psXs0ss0XqUq 2sin2ps+q jps+qj)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F3 11.955 Tf 9.3 0 Td[("ps+qs0=)]TJ /F7 11.955 Tf 10.49 8.09 Td[(m"ps psXs0ss0XqUq 2sin2ps+q jps+qj)]TJ /F7 11.955 Tf 10.5 8.09 Td[(psq mcos)]TJ /F7 11.955 Tf 15.42 8.09 Td[(q2 2m)]TJ /F7 11.955 Tf 11.96 0 Td[(s0jps+qj+sps. (A)Atermofordercanbeobtainedbyexpandingthe-functionwithrespectto2s=m2"ps p2ssXs0XqUq 2q2sin2 jps+qj2pFq mcos+q2 2m="ps 2vFs vFZ2pF0dqUqq2 2p2Fs 1)]TJ /F7 11.955 Tf 16.84 8.08 Td[(q2 4p2F. (A)ComparingEqs.( A )and( A ),weseethats=1s+2s=0toorder,whichisanagreementwiththepreviousresults.[ 48 55 ]IntheCoulombcase,Refs.[ 48 66 ]alsondanrs2lncorrectiontothemass,whichisthesamefortwoRashbasubbands.ThiscorrectionisproducedbythesecondterminEq.( A ).Toseethis,onecankeeponlythelogarithmicintegralsin( A ).Differentiating( A )twicewithrespectto,weobtain @2s @2=)]TJ /F7 11.955 Tf 10.49 8.09 Td[(e2"ps v3FZ2mvF2mjjdq q=)]TJ /F7 11.955 Tf 10.49 8.09 Td[(e2"ps v3FlnvF .(A)IntegratingEq.( A )over,wereproducetheresultofRef.[ 48 66 ].Inaddition,Ref.[ 66 ]obtainsvelocitysplittingassrs3ln.Suchatermisproducedbyboth1sand2s.Indeed,keepingagainonlylogarithmicintegralsanddifferentiating( A )threetimeswithrespecttogives @31s @3=)]TJ /F7 11.955 Tf 9.3 0 Td[(s3e2"ps v4FZ2mvF2mdq q=)]TJ /F7 11.955 Tf 9.3 0 Td[(s3e2"ps v4FlnvF .(A) 74

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Thereisalsoanothercontributionofthesameordercomingfrom2s.ItfollowsfromEq.( A )that2s=)]TJ /F7 11.955 Tf 14.84 8.09 Td[(m"ps 82ps"Z2ps0qdqU(q)Z)]TJ /F21 5.978 Tf 11.07 3.53 Td[(q 2ps)]TJ /F6 7.97 Tf 6.58 0 Td[(1d(cos)q2sin (p2s+q2+2qpscos)3=2)]TJ /F11 11.955 Tf 11.29 16.28 Td[(Z2p)]TJ /F21 5.978 Tf 5.76 0 Td[(s4mjjqdqU(q)Z)]TJ /F21 5.978 Tf 11.07 3.53 Td[(q 2ps+4ms q)]TJ /F6 7.97 Tf 6.59 0 Td[(1d(cos)q2sin (p2s+q2+2qpscos)3=2#. (A)Keepingagainonlylogarithmicintegrals,wend2s=m"ps 162psZ2p)]TJ /F21 5.978 Tf 5.75 0 Td[(s4mjjq3 p3sdqU(q)s 1)]TJ /F11 11.955 Tf 11.95 16.86 Td[(q 2ps)]TJ /F5 11.955 Tf 13.15 8.09 Td[(4ms q24ms q)]TJ /F7 11.955 Tf 18.27 8.09 Td[(q 2ps. (A)Differentiating( A )threetimeswithrespecttogives @32s @3=)]TJ /F7 11.955 Tf 9.3 0 Td[(s4e2"ps v4FZ2pF4mdq q=)]TJ /F7 11.955 Tf 9.3 0 Td[(s4e2"ps v4FlnvF .(A)CombiningEqs.( A )and( A ),weobtain @3s @3=)]TJ /F7 11.955 Tf 9.3 0 Td[(s7e2"ps v4FlnvF ,(A)which,afterintegrationover,reproducesthemainlogarithmictermintheresultofRef.[ 66 ].[Webelievethataconstantinsidethelogarithm,alsoobtainedinRef.[ 66 ],exceedstheoverallaccuracyofthecalculation.]NotethatthevelocitysplittingfrombothcontributionsinEqs.( A )and( A )isaresultofinter-bandtransitionsonly. 75

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APPENDIXBOUT-OF-PLANESPINSUSCEPTIBILITYOFNON-INTERACTINGRASHBAELECTRONS:ATHERMODYNAMICCALCULATIONInthisAppendix,weshowthattheout-of-planespinsusceptibility,zz,ofanon-interactingelectrongaswithRashbaSOcouplingisdeterminedbythestatesinbetweenfromthetwospin-splitFSs.Sincewearedealingonlywithfreestateshere,theindexfwillbesuppressedwhilequantitiesintheabsenceofthemagneticeldwillbedenotedbythesuperscript0.InthepresenceofaweakmagneticeldHinthez-direction,theelectronspectrumchangesto s(p)=0s(p)+s2 2p,(B)where0s(p)coincideswithEq.( 2 )and=gBH=2.Theground-stateenergyisgivenby E=XsZdpp 2s(p)(ps)]TJ /F7 11.955 Tf 11.95 0 Td[(p),(B)wheretheFermimomentaofthesubbandsarefoundfrom s(ps)=F.(B)Itiseasytocheckthat,atxednumberdensity,Fisnotaffectedbythemagneticeld.OnecanthusreplaceFby0s(p0s)inEq.( B ),whichgivesforthecorrectionstotheFermimomenta ps=p0s+ps=p0s)]TJ /F7 11.955 Tf 11.96 0 Td[(s2 2v0p0s,(B)whereistheFermivelocityineachofthesubbands,givenbyEq.( 2 ).Itisconvenienttosubtracttheeld-independentpartfromE,andsplittheremainderintotwopartsas E)]TJ /F7 11.955 Tf 11.96 0 Td[(E0=Eon+Eo,(B) 76

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whereEonisthecontributionfromthestatesneartheFSs Eon=1 2XsZp0s+psp0sdpp0s(p)(B)andEoisthecontributionfromthestatesawayfromtheFSs Eo=1 2XsZp0s0dpps2 2p.(B)TheFScontributionvanishestoorder2 Eon=1 2Xspsps0s(ps)=0.(B)Ontheotherhand,theoff-FScontributionbecomes Eo=1 2Zp0)]TJ /F4 7.97 Tf -9.69 -21.47 Td[(p0+dpp)]TJ /F5 11.955 Tf 14.03 8.09 Td[(2 2p=)]TJ /F7 11.955 Tf 10.49 8.09 Td[(m2 2,(B)whichgivesacorrectresult0zz=g22Bm=4(Ref.[ 44 ]).Therefore,theout-of-planespinsusceptibilityofnon-interactingRashbafermionscomesentirelyfromthestatesinbetweenthetwoFSs. 77

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APPENDIXCKOHNANOMALYINAFREEELECTRONGASWITHRASHBASPIN-ORBITCOUPLINGInthisAppendix,wederiveEqs.( 2 2 )fortheKohnanomaliesinthepolarizationbubbleofnon-interactingRashbafermions(Fig. C-1 ): (Q)=Xs,s0ss0(Q),(C)where ss0(Q)=1 2ZP[1+ss0cos(p)]TJ /F3 11.955 Tf 11.95 0 Td[(p+q)]gs(P)gs0(P+Q)=1 2Zd2p (2)2[1+ss0cos(p)]TJ /F3 11.955 Tf 11.96 0 Td[(p+q)]()]TJ /F3 11.955 Tf 9.3 0 Td[(s,p)1 i)]TJ /F3 11.955 Tf 11.96 0 Td[(s0,p+q+s,k)]TJ /F5 11.955 Tf 54.04 8.09 Td[(1 i+s,p)]TJ /F6 7.97 Tf 6.59 0 Td[(q)]TJ /F3 11.955 Tf 11.95 0 Td[(s0,p. (C) (AsinAppendices A and B ,wesuppressthesuperscriptfdenotingthepropertiesofaninteraction-freesystem.)Forqnear2pF,weexpandthedifferenceofthequasiparticleenergiesas p+qs)]TJ /F3 11.955 Tf 11.95 0 Td[(ps=)]TJ /F5 11.955 Tf 9.3 0 Td[(2ps+vFq0+vFpF2,(C)whereq0=q)]TJ /F5 11.955 Tf 12.17 0 Td[(2psandtheangle=\()]TJ /F14 11.955 Tf 9.3 0 Td[(p,q)istakentobesmall.ToleadingorderinSO,theprefactorsintheq0and2termscanbeandwerereplacedbytheirvaluesintheabsenceofSO.(ForthesecondterminEq.( C ),theangle=\(p,q)istakentobe FigureC-1. Polarizationbubble,givenbyEq. C ,with(2+1)momentumtransferQ. 78

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small.)UsingEq.( C ),replacing[1+cos(p)]TJ /F3 11.955 Tf 12.25 0 Td[(p+q)]by22,andintegratingoverpinEq.( C ),weobtainfor=0ands=s0 ss(q)= 2Zc02ln2+q0 pFd,(C)wherecisacutoffwhoseparticularchoicedoesnotaffectthesingularpartoftheq-dependence.Toextractthesingularpart,wedifferentiatess(q)twicewithrespecttoq,whichmakestheintegraltoconvergeinthelimitofc!1,andobtain ss(q)=ss(q=2ps))]TJ /F3 11.955 Tf 13.15 8.09 Td[( 6(q)]TJ /F5 11.955 Tf 11.95 0 Td[(2ps)q)]TJ /F5 11.955 Tf 11.96 0 Td[(2ps pF3=2.(C)The(q0)3=2anomalyinss(q)isweakerthanthesquare-rootanomalyintheabsenceofSO.ThisisaconsequenceofthefactthatbackscatteringwithinthesameRashbasubbandisforbidden,whichismanifestedbyasmallfactorof2intheintegrandofEq.( C ).Asimilarprocedureisappliedtos,)]TJ /F4 7.97 Tf 6.59 0 Td[(s,whichresultsfrombackscatteringbetweendifferentsubbandsandhencedoesnothaveasmallfactorof2.Withq0replacednowbyq)]TJ /F5 11.955 Tf 11.95 0 Td[(2pF,weobtaintheusualsquare-rootKohnanomaly s,)]TJ /F4 7.97 Tf 6.59 0 Td[(s(q)=s,)]TJ /F4 7.97 Tf 6.58 0 Td[(s(q=2pF)+ 2(q)]TJ /F5 11.955 Tf 11.96 0 Td[(2pF)q)]TJ /F5 11.955 Tf 11.95 0 Td[(2pF pF1=2.(C)Notethatwhenthebubbleisinsertedintotheinteractionvertex,asinFig. 2-2 ,themaximummomentumenteringthebubbleis2p)]TJ /F1 11.955 Tf 10.4 1.8 Td[((whenboththeincomingelectronsarethe)]TJ /F1 11.955 Tf 12.62 0 Td[(subband).Therefore,thesingularityin\000,presentonlyforq>2p)]TJ /F1 11.955 Tf 7.08 1.79 Td[(,isoutsidetherangeofallowedmomenta.Withthisinmind,thenon-analyticpartof,relevantforthecalculationoftheLandaufunctioninSec. 2.5.3 ,canbewrittenasinEqs.( 2 2 )ofthemaintext. 79

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APPENDIXDNON-ANALYTICCONTRIBUTIONSTOTHELANDAUFUNCTIONInthisAppendix,wecalculateO(s2U2)termsintheLandaufunctionproducedbyverticesbanddinFig. 2-2 .AcontributionfromdiagramciscalculatedinthemaintextofSec. 2.5.3 .Diagramdreads )]TJ /F4 7.97 Tf 6.78 4.94 Td[(dss0=)]TJ /F7 11.955 Tf 10.5 8.09 Td[(U2 4Xr,t,L[1+srcos(p)]TJ /F3 11.955 Tf 11.96 0 Td[(l)][1+s0tcos(p0)]TJ /F3 11.955 Tf 11.96 0 Td[(l0)]gr(L)gt(L0), (D) whereL0=(l+p0)]TJ /F14 11.955 Tf 11.96 0 Td[(p,!l).Anon-analyticpartof)]TJ /F4 7.97 Tf 6.77 4.33 Td[(dss0comesfrombackscatteringprocesseswithp)]TJ /F14 11.955 Tf 22.89 0 Td[(p0.Therefore,internalmomentamustbechosensuchthatlpandl0p0)]TJ /F14 11.955 Tf 21.92 0 Td[(p,andbothcosinetermsinEq.( D )arenear+1.(Themagnitudesoftheinternalmomentamaydifferfromexternalones,buttakingthiseffectintoaccountisnotnecessarytolowestorderin.)Thes=s0=+1channeldoesnotcontainasingularity,becausethemaximummomentumtransferinthischannelis2p+,whileanyconvolutionoftwoGreen'sfunctionsdoesnotdependonthedifferenceoftheirmomentaforq2p+(cf.Appendix C ).Wethusfocusonthes=s0=)]TJ /F5 11.955 Tf 9.3 0 Td[(1ands=)]TJ /F7 11.955 Tf 9.3 0 Td[(s0channels.Inthes=s0=)]TJ /F5 11.955 Tf 9.3 0 Td[(1channel,theangular-dependentfactorsaremaximalforr=t=)]TJ /F5 11.955 Tf 9.3 0 Td[(1;however,thesingularityintheconvolutionoftwog)]TJ /F1 11.955 Tf 10.4 1.8 Td[(functionisoutsidetheallowedmomentumtransferrange.Thenextbestchoiceisr=)]TJ /F7 11.955 Tf 9.3 0 Td[(t,whenoneofangular-dependentfactorsissmallbutthesecondoneislarge.Thechoicet=r=1makesbothangular-dependentfactorstobesmallandisdiscarded.Withallanglesmeasuredfromp(p=0),wedenote0p=)]TJ /F3 11.955 Tf 12.21 0 Td[(,l=,andl0=)]TJ /F3 11.955 Tf 12.2 0 Td[(0andtake,,and0tosmall.Aftersomeelementarygeometry,wendthat0=+,whichimpliesthattheangular-dependentfactorsinfrontoftheg+g)]TJ /F1 11.955 Tf 10.41 1.79 Td[(andg)]TJ /F7 11.955 Tf 7.09 1.79 Td[(g+combinationsarethesame.Therefore, )]TJ /F4 7.97 Tf 6.78 4.94 Td[(d\000=)]TJ /F7 11.955 Tf 10.49 8.09 Td[(U2 2ZL2g+(L)g)]TJ /F5 11.955 Tf 7.09 1.79 Td[((L0),(D) 80

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andsimilarly )]TJ /F4 7.97 Tf 6.78 4.93 Td[(d+)]TJ /F5 11.955 Tf 10.41 2.96 Td[(=)]TJ /F7 11.955 Tf 10.49 8.08 Td[(U2 4ZL2g+(L)g+(L0).(D)Theintegralsintheequationsabovearethesameasforthe++componentofthepolarizationbubble[cf.Eq.( C ],exceptforthatthemomentumtransferin)]TJ /F4 7.97 Tf 6.78 4.33 Td[(d\000ismeasuredfrom2pFratherthanfrom2p+.Withthisdifferencetakenintoaccount,weobtaintheresultsfor)]TJ /F4 7.97 Tf 6.77 4.34 Td[(d\000and)]TJ /F4 7.97 Tf 6.78 4.34 Td[(d+)]TJ /F1 11.955 Tf 10.41 2.96 Td[(inEqs.( 2 )and( 2 )ofthemaintext.Thetwowine-glassdiagramsb)areequaltoeachother,andtheirsumisgivenby )]TJ /F4 7.97 Tf 6.77 4.94 Td[(bss0=U2 8Xr,tZL[1)]TJ /F7 11.955 Tf 11.96 0 Td[(ss0ei(p0)]TJ /F10 7.97 Tf 6.59 0 Td[(p)][1+rtei(l)]TJ /F10 7.97 Tf 6.59 0 Td[(l0)][1+srei(p)]TJ /F10 7.97 Tf 6.59 0 Td[(l)][1+s0tei(l0)]TJ /F10 7.97 Tf 6.59 0 Td[(p0)]gr(L)gt(L0). (D) Again,wehavelpandl0p0)]TJ /F14 11.955 Tf 21.92 0 Td[(p,sothatr=)]TJ /F7 11.955 Tf 9.3 0 Td[(tforthes=s0=)]TJ /F5 11.955 Tf 9.3 0 Td[(1channel.Usingthedenitionsofsmallanglesabove,weobtain )]TJ /F4 7.97 Tf 6.78 4.93 Td[(b\000=U2 2)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F7 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(iZL)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(1+ei(2+))]TJ /F5 11.955 Tf 12.95 -9.68 Td[(1)]TJ /F7 11.955 Tf 11.95 0 Td[(e)]TJ /F6 7.97 Tf 6.58 0 Td[(2i[g+(L)g)]TJ /F5 11.955 Tf 7.09 1.8 Td[((L0)+g)]TJ /F5 11.955 Tf 7.09 1.8 Td[((L)g+(L0)]. (D) Expandingtheinternalangular-dependentfactorstorstorderinand,weobtainanintegralthatvanishesbyparity.The2termisnitebutstillcomeswithasmallfactoroffromtheexternalmatrixelement.Therefore,)]TJ /F4 7.97 Tf 6.78 4.34 Td[(b\000doesnotcontributetoorder2.Forthes=)]TJ /F7 11.955 Tf 9.3 0 Td[(s0channel,onlythecombinationr=t=+1contributes.Followingthesamestepsasfor)]TJ /F4 7.97 Tf 6.78 4.34 Td[(b\000,wearriveat )]TJ /F4 7.97 Tf 6.78 4.94 Td[(b+)]TJ /F5 11.955 Tf 10.41 2.96 Td[(=U2 8)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(1+e)]TJ /F4 7.97 Tf 6.59 0 Td[(iZL)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F7 11.955 Tf 11.95 0 Td[(ei(2+))]TJ /F5 11.955 Tf 12.95 -9.68 Td[(1)]TJ /F7 11.955 Tf 11.96 0 Td[(e)]TJ /F6 7.97 Tf 6.59 0 Td[(2ig+(L)g+(L0).(D)Expandingtheangularfactorstolowestorder,weobtain )]TJ /F4 7.97 Tf 6.77 4.94 Td[(b+)]TJ /F5 11.955 Tf 10.4 2.95 Td[(=U2ZL2g+(L)g+(L0),(D)whichcoincideswiththeintegralforssinEq.( C ).Combiningeverythingtogether,weobtaintheverticesinEqs.( 2 )and( 2 )ofthemaintext. 81

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BIOGRAPHICALSKETCH AliAshrawasbornin1985inTehran,Iran.HereceivedhishighschooldiplomainphysicsandmathematicsfromKamalhighschoolin2002,andhisB.Sc.inAtomic,Molecular,andOpticsphysicsfromUniversityofTehranin2008.Duringhisundergraduatestudies,heworkedonaseveraltopicsfrommathematicalphysicsandalternativequantumtheoriestoappliedphysicsandoptics.HearrivedattheUniversityofFlorida(UF),Gainesville,FL,infall2008topursuehisPh.D.inphysics,whereherstworkedonmathematicalphysicsofnon-renormalizablequantumeldtheoriesandquantumgravitywithProf.JohnKlauder,andthenjoinedthegroupofProf.DmitriiMaslovtoworkintheeldoftheoreticalcondensedmatterphysics.HereceivedhisPh.D.inphysicsfromUFinAugust2013. 86