The Quantum Hall Effect Unified Scaling Theory and Quasi-particles at the Edge

TheQuantumHallEffect

UnifiedScalingTheoryandQuasi-particlesattheEdge

A.M.M.PruiskenandK.Schoutens

InstituteforTheoreticalPhysics

UniversityofAmsterdamValckenierstraat651018XEAmsterdamTHENETHERLANDS

Abstract

WeaddresstwofundamentalissuesinthephysicsofthequantumHalleffect:aunifieddescriptionofscalingbehaviorofconductancesintheintegralandfractionalregimes,andaquasi-particleformulationofthechiralLuttingerLiquidsthatdescribethedynamicsofedgeexcitationsinthefractionalregime.

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1Introductionandsummary

AunifieddescriptionofthescalingbehaviorintheintegralandfractionalquantumHallregimesrequiresatreatmentthatcombinestheeffectsofimpuritiesandofelectron-electroninteractions.Therearetworecentdevelopmentsthathaveclearedthewayforaseriouseffortinthisdirection.ThefirstistheuseofChern-Simonsgaugetheory,whichmakesitpossibletoconstructamappingbetweentheintegralandfractionalregimes.TheseconddevelopmentistherecentanalysisbyPruiskenandBaranov(1995)ofscalingbehaviorinthepresenceoftheCoulombinteraction.Inthispaperwebrieflyreviewtheseresults.

EffectivetheoriesforexcitationsattheedgeofaquantumHallsampletaketheformofachiralFermiliquid(intheintegralregime)orchiralLuttingerliquid(inthefractionalregime).TheLuttingerliquidbehaviorofedgecurrentsinquantumHallsampleswithfillingfractionν=1/3hasbeenprobedinrecentexperiments(Millikenetal1996,Changetal1996).Inrecenttheoreticalwork,theConformalFieldTheoriesthatdescribethechiralLuttingerliquidsforfillingfractionsν=1/(2m+1)havebeenanalyzedusingaso-calledquasi-particleformulation.WeshallherebrieflyexplainthatmanyoftheremarkablefeaturesofthefractionalquantumHalledgedynamicsarenaturallyexplainedinasuchaquasi-particlepicture.

2Thequestforaunifiedscalingtheory

OurmicroscopicunderstandingofthequantumHalleffect(qHe)haslargelydevel-opedalongtwoseparatepathwayswhichforalongtimeappearedtohaveverylittleincommon(PrangeandGirvin1990).Thefirstandpossiblymostpopularlystudiedroute—initiatedbyLaughlin—isthatofthe“clean”statesofthe2Delectrongas.Thefocusisprimarilyontheeffectsofstrongcorrelationbetweentheelectrons.Thefrac-tionalqHe,whichmanifestsitselfonlyinhighquality,highmobilityheterostructures(Chang1990),isgenerallybelievedtobesuchastronglycorrelatedphenomenonwithnovelfeaturessuchasfractionalstatisticsandchargeofquasi-particles.

Thesecondapproach—whichsofarhasbeenrestrictedtotheregimeoftheinte-gralqHe—isthatofthe“impure”ordisorderdominatedstatesofthe2Delectrongas.Here,themainphysicalobjectiveistounderstandthephenomenonofAndersonlocal-izationoffreeelectrons,whichmanifestsitselfmacroscopicallythrough“scaling”oftheconductanceswithvaryingexperimentalparameterssuchasmagneticfieldandtem-perature(Pruisken1988).Theso-called“scalingtheory”oftheintegralqHe(Fig.1)hasgeneratedasubstantialamountofexperimental(Weietal1988,Kochetal1991,Engeletal1993,Hwangetal1993)andnumerical(Huckensteinetal1990,Huoet

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Figure1:ScalingdiagramoftheconductancesintheintegralquantumHallregime(Pruisken

1988).Thearrowsindicatethescalingtowardlargesamplesizes(orlowT)andtheflowlinesareperiodicinσxywithperiode2/h.

al1993)resultsonthecriticalaspectsofthe“transitions”betweenadjacentquantumHallplateaus.Criticalityplaysafundamentalroleinthetheoryofmetalsandinsula-torsingeneralandhereitelucidatesthesurprisingmechanismof“delocalization”ofthe2Delectrongasinstrongmagneticfields(PhysicsToday1988).

Despitethefactthattheextremetheoreticalapproachesof“clean”and“disor-dered”stateswereoriginallyformulatedinatotallydifferentlanguage,itisnaturaltoexpectthatthetrue,experimentalsituationmustvarycontinuouslybetweentheseextremes,theprecisephysicaloutcomebeingdeterminedbysamplespecificparame-terssuchasamountandtypeofdisorderandtemperature.Severalauthors(Laughlinetal1985)haveelaboratedonanimportantpredictionoftherenormalizationtheoryoftheintegralqHewhichsaysthatthelowerportionoftheσxx−σxyscalingdiagramisinfacta“forbidden”regionforfreeelectrons.Thisforbiddenregion,indicatedby“?”inFig.1,wassubsequentlyrecognizedasthefractionalquantumHallregime.Hence,therehasbeenalongstandingexpectationofa“unifiedscalingtheory”whichsimultaneouslyincorporatestheeffectsofstrongcorrelationanddisorder,i.e.describesbothintegralandfractionalqHe.

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Chern-Simonsgaugetheoryandinstantonvac-uum

InthisSectionwediscusstwoimportantrecentdevelopmentswhichinprinciplecanbe(andhavealreadybeen)usedinordertosubstantiatetheabovementionedconjec-tureofaunifiedscalingtheory.First,thereistheChern-Simons(CS)gaugetheoryapproach(Wilczek1982,Zhangetal19,LopezandFradkin1991,Kivelsonetal1992,Halperinetal1993)whichhasbecomeinstrumentalalsoinotherstronglycor-relatedelectronsystemssuchasthechiralspinliquidandanyonsuperconductivity.Theseconddevelopmentisarecentanalysisofscalingbehaviorinthepresenceofelectron-electroninteractions(PruiskenandBaranov1995).

(i)SeveralauthorshaveexploitedthefactthatbycouplingCSgaugefieldstoan

electroniccurrentdensityoneproducesatheorywithprobabilityamplitudeswhichareidenticaltothoseobtainedintheoriginaltheory(i.e.withoutthegaugefieldspresent),providedtheCScouplingconstantor“statisticsangle”θischosentobeamultipleof2π.

Theinterestingobservationisbeingmade(LopezandFradkin1991)thatthestatisticalgaugefields—atameanfieldlevel—produceapicturewhichinallrespectsisidenticaltothe“compositefermion”theoryofJain.Withinthesemiclassicalapproximation(i.e.meanfieldplusgaussianfluctuations),however,allthephysicsoftheLaughlinincompressibleliquidstateisreproduced.Inthisway,CSgaugetheoryformallyprovidesa”mapping”betweentheintegralandfractionalquantumHallplateaus.

AlthoughthesemiclassicaltheoryoftheCSgaugefieldsisstrictlyvalidonlyifoneassumesagapintheenergyspectrum,thesameprocedurehasneverthe-lessbeenappliedtoanypossiblecompressible(metallic)stateintheproblem(Kivelsonetal1992).Ifsuchanapproachcanindeedbejustifiedingeneral,itwouldimplythatthescalingbehavioroftheintegralregimecanformallybemappedontothatofthefractionalregime.Anexplicitsl(2,Z)dualsymmetryhasbeenobtainedbetweentheintegralquantumHallregime,wheretheresultsofthescalingtheoryapply,andthefree-electron-forbiddenfractionalquantumHallregime,denotedby“?”inFig.1.

Unfortunately,thereexistsverylittleknowledgewhichwouldjustifythevalidityofsemiclassicalapproximationsforcompressibleormetallicstates.Forinstance,althoughthehalf-integralquantumHalleffectformallyappearswithintheframe-workofthesemiclassicaltheory,itisknown(Halperinetal1993)thatthegauge

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fieldfluctuationsgiverisetodivergentcontributionstothequasi-particlepropa-gator,indicatingthatitisdifficulttoextract(eitherexplicitorimplicit)knowl-edgeabouttheinfraredbehaviorofthetheory.Another,possiblymoreseriouscomplicationarisesfromthefactthattheconductancefluctuationsareverylargeatthequantumHalltransitions(CohenandPruisken1993,1994).ThisraisesfundamentalquestionsaboutthevalidityoflinearresponsetheoryasawholeandofsemiclassicalanalysesoftheCSgaugetheoryinparticular.Amicroscopictheoryforelectronicdisorderisobviouslywhatisneededinordertocompletelyunderstandtheproblemsathandand,ultimately,totakefulladvantageoftheCSgaugetheoryapproach.

Muchoftheproblematics,raisedinthisSection,hasalreadybeenencounteredmoreexplicitlyintheextensiveexperimentalstudiesonscalingintheintegralquantumHallregime(PruiskenandWei1993).Thatis,detailedcomparisonbetweenthepredictionsofthefreeelectronrenormalizationtheoryandtheex-perimentaldatatakenfromlowmobilityheterostructureshasprovidedimpor-tantinsightinthepossibleroleplayedbytheelectron-electroninteractionsintheproblem.

(ii)ThisbringsusbacktooneofthelongstandingproblemsinthetheoryoftheqHe,

whichiswhetherandhowthetopologicalconceptofaninstantonvacuum—whichhassuchadramaticimpactonthelocalizationoffreeelectrons(Pruisken1984,1987)—hasanyrelevanceforasystemofinteractingelectrons.TheissuewasaddressedinarecentdetailedanalysisbyPruiskenandBaranov(1995).Finkelstein’sgeneralizedsigmamodeltheory(Finkelstein1983,1984,1994)wasusedasastartingpointfortheanalysisandadaptedtotheproblemofstrongLandaulevelquantization.Ithasturnedoutthatinstantoneffectsgiverisetoessentiallythesamescalingdiagramfortheconductancesaswasobtainedwithinthefreeelectrontheory,withoneimportantexception:thetemperaturealsoscalesnon-trivially,andthisimpliesanon-trivialdynamicalscalingbehaviorintheproblem.

4Towardaunifiedscalingtheory

AlthoughtheCoulombinteractionproblemturnsouttosharemanyfeatureswiththefreeelectrontheoryoftheqHe(asymptoticfreedom,instantonsetc.),itistruethatmanyofthephysicalobjectivesoftheoriginal,perturbativeFinkelsteinapproachhaveactuallyremainedwithoutananswer.Foronething,itisunclearhowoneshouldgobeyondtheperturbativemetallicregimeandextendthetheorytoinclude

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Figure2:ScalingdiagramforthefractionalquantumHallregimewithσxy<

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tronicspecificheatCV,whichrevealsmuchofthelowenergyphysicsoftheinteracting,disorderedelectrongasneartwospatialdimensions.ThefollowingexpressionfortheenergyE(relativetothezeropointenergy)hasbeenextractedfromweakcoupling,perturbativeexpansions

E=

󰀁

∞0

dǫǫnbe(T)ρqp(ǫ).(4.1)

Themeaningofthesymbolsisasfollows.Theintegralisovertheexcitationenergiesrelativetothechemicalpotential.Thenbe=1

thesigmamodeleffectiveaction,provideinprincipleaunifyingrenormalizationgrouptheoryofintegralandfractionalregimes.Fig.2illustratesthescalingresultsobtainedfromweakcoupling(perturbativeandnon-perturbative)analyses.ItisinterestingtonoticethatFig.2differsfromthehistoricalguess(Laughlinetal1985)byaslightredefinitionofthefixedpointsontheevendenominatorlines1/2,1/4,etc.Thedifferenceisfundamental,however,sincetheunstablefixedpointsatσxy=1/2,1/4etc.nowdescribethehalfintegerphase(Halperinetal1993)whichservesasaperturbative,weakcouplingregimeinthepresentcontext.

OurunifyingtheoryfortheintegerandfractionalquantumHallregimesimpliesthatthereisthehighlynon-trivialrelationbetween”edge”and”bulk”effectsandthisisalargelyopenproblemasofyet.Interestinglyenough,onehastodealherewithtwotopologicalconceptssimultaneously(CSgaugetheoryandinstantonvacuum)whichtogethershouldrelatethedisorderedquasi-particlesystemofthebulktotheLuttingerliquidbehavioroftheedgestates(seeSection5below).

5Quasi-particlesattheedge

Inthepastfiveyears,ithasbeenrecognizedthatmuchoftheunusualphysicsofthe(fractionalorinteger)quantumHalleffectisreflectedbythepropertiesoftheso-callededgestates,whicharelocalizedatthephysicalboundaryofaquantumHallsample.Whiletheedgestatesformaone-dimensionalFermiliquidforintegerfillingfrac-tions,theyshownon-Fermiliquidpropertiesinthefractionalregime.Thetheoreticalprediction(Wen1990,1992)isthattheedgestatesformachiralLuttingerLiquidwithcharacteristicparametersdependingonthefillingfractionν.TheseparametersaredirectlyrelatedtotheChernSimonsgaugetheoryforthebulkdegreesoffreedom(seeSection3).Theadjective“chiral”meansthatatagivenboundarytheedgeexcitationscanonlytravelinonedirection.Duetothis,backscatteringisnotpossible.ThechiralLuttingerLiquidhasbeenpredictedtobestableanduniversal.

Toactuallytestthesetheoreticalpredictions,oneneedsanexperimentthatdirectlyprobesthenon-trivialpropertiesoftheedgestates.Onepossibilityforthisistocreateaso-calledpointcontactbetweentwofractionalquantumHalledgesandtostudythetunnelingoftheedgeexcitationsthroughthiscontactasafunctionoftemperatureandofanappliedgatevoltage.Forfillingfractionν=1/3,thisexperimenthasbeenperformed(Millikenetal1996)andtheexperimentalresultshavebeenclaimedtobeinagreementwiththeoreticalpredictions.Inanotherexperiment(Changetal1996)theI−Vcharacteristicsforacurrenttunnelingfromabulkmetalintoaν=1/3edgehavebeenmeasured.Inthenon-linearregime,thedatacanbefittedtoapowerlawI∝Vαwithα=2.7±.06foraspecificsample.

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ThechiralLuttingerliquidsforthefractionalqHeedgedynamicsareexamplesofso-calledConformalFieldTheories.Forthespecificcaseofedgetheoriesforfillingfractionν=1/(2m+1),m=1,2,...,thisConformalFieldTheoryisparticularlysimple.Itisusuallyanalyzedin“bosonized”form,theorganizingprinciplebeingtheU(1)Kac-Moodyalgebrageneratedbythechargedensityoperator.Wehereproposethatitismuchmorenaturaltoanalyzethistheorydirectlyintermsofthebasicquasi-particles,whicharenotelectronsbutquasi-particlesofchargee/(2m+1)andofscalingdimensionx=1/(4m+2).Theedgeelectronisacompositeof(2m+1)ofthesequasi-particlesandassuchithasscalingdimension(2m+1)2x=(2m+1)/2.Thesystematicsofthequasi-particleformulationoftheedgetheoryforν=1/(2m+1)haverecentlybeenworkedout(Iso1995,SchoutensandvanElburg).Theyarecloselyanalogoustothesystematicsoftheso-calledspinonformulationsoftheConformalFieldTheoryfortheHeisenbergandHaldane-Shastryspinchains(Haldaneetal1992,Bouwknegtetal1994),whichisformallyidenticaltothecasem=1/2oftheqHeedgetheories.

Toillustrateourpointofview,wegiveonesimplebutimportantexample,whichisthetunnelingdensityofstatesA(ǫ).Thisquantityequalsthespectraldensityforone-electronexcitationswithexcessenergyǫoverthegroundstate.InfreeelectrontheoriesA(ǫ)doesnotdependonǫ.However,letusnowconsidercreatingaone-electronexcitationinatheorywithν=1/(2m+1).Thismeanscreating(2m+1)quasi-particleswiththesingleconstraintthatthesumoftheirenergiesequalsǫ.Clearly,anexpressionforA(ǫ)willcontain2mfreeintegrationsoverenergyvariables.ItwillthereforedependonǫasA(ǫ)∝(ǫ−µ)2m.Ourreasoninghere,whichconfirmstheresultof(Wen1990),nicelyillustratesthepowerofthequasi-particleformulation.ThepowerlawforA(ǫ)impliesnon-linearI−Vcharacteristicsforelectrontun-neling.Forν=1/3thequantitativepredictionisI∝V3,whichcanbecomparedwiththeChangetalexperiment.

Theexacttheoreticalanalysis(Fendleyetal1995)oftheuniversalconductancecurvefortheMillikenetalexperimentmadeuseofanotherquasi-particlebasisoftheν=1/3edgetheory.Thisbasishasitsoriginintheoreticalwork(GoshalandZamolodchikov1994)onintegrableboundaryscattering.Thepreciserelationamongthevariousquasi-particleformulationsthatarerelevantforthetunnelingexperimentsiscurrentlyunderstudy.

Acknowledgement.

ThisworkwassupportedinpartbythefoundationFOMoftheNetherlands.

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