girvin,introduction to the fractional quantum hall effect

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1、TheQuantumHallEffect,133–162
cBirkh¨auserVerlag,Basel,2005Poincar´eSeminar2004IntroductiontotheFractionalQuantumHallEffectStevenM.Girvin1IntroductionThequantumHalleffect(QHE)isoneofthemostremarkablecondensed-matterphenomenadiscoveredinthesecondhalfofthe20thcentury.Itrivalssupercon-ductivityinit

2、sfundamentalsignificanceasamanifestationofquantummechanicsonmacroscopicscales.Thebasicexperimentalobservationforatwo-dimensionalelectrongassubjectedtoastrongmagneticfieldisnearlyvanishingdissipationσxx→0(1)andspecialvaluesoftheHallconductancee2σxy=ν(2)hgivenbythequantumofelectricalconductance(

3、e2/h)multipliedbyaquantumnumberν.Thisquantizationisuniversalandindependentofallmicroscopicdetailssuchasthetypeofsemiconductormaterial,thepurityofthesample,theprecisevalueofthemagneticfield,andsoforth.Asaresult,theeffectisnowusedtomaintain(butnotdefine)thestandardofelectricalresistancebymetrolog

4、ylabo-ratoriesaroundtheworld.Inaddition,sincethespeedoflightisnowdefined,ameasurementofe2/hisequivalenttoameasurementofthefinestructureconstantoffundamentalimportanceinquantumelectrodynamics.Fig.(1)showstheremarkabletransportdataforarealdeviceinthequan-tumHallregime.InsteadofaHallresistivitywh

5、ichissimplyalinearfunctionofmagneticfield,weseeaseriesofso-calledHallplateausinwhichρxyisauniversalconstant1hρxy=−(3)νe2independentofallmicroscopicdetails(includingtheprecisevalueofthemagneticfield).Associatedwitheachoftheseplateausisadramaticdecreaseinthedissi-pativeresistivityρxx−→0whichdrop

6、sasmuchas13ordersofmagnitudeintheplateauregions.Clearlythesystemisundergoingsomesortofsequenceofphasetransitionsintohighlyidealizeddissipationlessstates.Justasinasuperconductor,thedissipationlessstatesupportspersistentcurrents.Intheso-calledintegerquantumHalleffect(IQHE)discoveredbyvonKl-itzi

7、ngin1980,thequantumnumberνisasimpleintegerwithaprecisionof134StevenM.GirvinFigure1:IntegerandfractionalquantumHalltransportdatashowingtheplateauregionsintheHallresistanceRHandassociateddipsinthedissipativeresistanceR.ThenumbersindicatetheLandauleve