2.9 Landau Levels and Quasiparticles in Magnetic Field
2.9.4 The Fractional Quantum Hall Effect and Higher-Order Quasiparticles
Theintegerquantum Halleffectis perhapsnottoosurprising.Experimentalists working with2DEGsystemsweresurprisedtoobserve,however,thatastheyincreasedthemagnetic fieldevenfurther,at verylowtemperaturetheysawplateauscorrespondingtoonlyone- thirdofthestatesinaLandaulevelbeingfilled,oralternatively,toafullLandaulevelof particleswithcharge of e/3.Just astheorginalLandaulevelseries correspondedtothe numberofstates N/A = νeB/h,where ν = 1,2,3,...,iftheexperimentsaredonewith highmagnetic fieldand high resolution,as shown in Figure2.42, anew Landauseries correspondingto ν = 13,23,33(= 1),43, ... isobserved. Plateauscorresponding toseries ofotherodd-integer fractionalcharges,suchas 15 and 17,arealsoobserved.Howcanwe understandthis?
AfulltreatmentofthefractionalquantumHalleffectrequiresunderstandingthemany- bodywavefunctionoftheelectrons;forageneralreviewseeLaughlin(1999).Wecanget abasicunderstandingofthiseffect,though,bythinkingintermsoftheinteractionsofthe electrons.
Supposethatwewanttowritedownthecorrectwavefunctionfortwoelectrons,taking intoaccountthefactthatthestatemustsatisfytheFermi–Diracstatisticsoftheelectrons
0
0 10 20 30
0.5 1 1.5 2 2.5 3
RH[h/e2]RL
Magnetic field (tesla) 1
4/3
7/5 5/3
4/5 5/7
2/3 3/5 4/7
5/9 4/9
3/7 2/5
3 2 1/3 4
±Fig.2.42 HallresistanceandlongitudinalresistancemeasuredforthefractionalquantumHalleffect,asafunctionof B,for constantcurrent.FromStormerandTsui(1997).
(thatis,mustchangesignoninterchangeoftheelectrons)andmustbeaneigenstateofthe totalangularmomentumoperator,
Lz=ẳ
i
−i±
´ xi ∂
∂yi − yi ∂
∂xi
à
. (2.9.35)
Theonlyformofwavefunctionthatsatisfiesbothconditionsis
ψ(z1, z2) ∝ (z1− z2)l, (2.9.36) where zn= xn+iyn = rneiθnisacomplexnumbercorrespondingtothepositionofelectron n intheplane,and l isanoddnumber.Ifweset z1 = 0,thenwehave
ψ(0, z) ∝ rleilθ. (2.9.37)
Thismeansthatthewavefunctionofanelectron,relativetoanotherelectron,musthave l zeros in the azimuthal direction. As discussed in Section 2.9.1, the phase change of the electroncorresponds to thenumber of flux quanta contained in itsorbit. Therefore, an electron in a state with l = 3 corresponds to an electron orbit confiningthree flux quanta.
Inafullmany-bodycalculationofthewavefunctionof N electrons,wemustminimize theenergyforawavefunctionwiththeform
ψ(z1, ... , zN) ∝ ÀN n<m
(zn− zm)l. (2.9.38) Withoutdelvingintothiscalculation,onecanalreadyseethatthegroundstatewillbeone inwhicheveryelectronisboundto l fluxquanta,where l isanoddinteger.Sincewemust putenergyintothesystemtotearawayafluxquantumfromanelectron,wecanviewthe fluxquanta aseffectivelypositivelychargedparticleswhichareattractedtothenegative chargeoftheelectron. Sincethestable conditionis l fluxquantaperelectron,each flux quantumwill haveaneffectivecharge of +e/l.Forexample, threefluxquanta boundto one electroncorrespondstoeach fluxquantum having +e/3charge.Inotherwords,we canaccountfortheinteractionoftheelectronsbywritingaHamiltonianwithaneffective chargeforthefluxquanta.
We therefore can go to a new quasiparticle picture. When the electron is bound to three flux quanta, it has two extra flux quanta compared to the one flux quantum that isalwaysassociatedwithanelectroninitsLandauorbit.Justas excitonsarecomposite quasiparticles consistingofanelectron anda holeboundtogether,we candefineanew composite fermionasaboundstateofanelectronandtwofluxquanta,withatotalcharge of −e + 23e = −13e.Thisnew quasiparticlecanthenbeseenas confiningasingleflux quantuminitslowestavailableLandauorbit,whichisdeterminedusingtheneweffective charge.3
3 It is also possible to think of the quasiparticle as having the same charge e but having “absorbed” two flux quanta, so that the effective magnetic field it feels is one-third of the original. This picture gives the same results for the Hall plateaus. See Jain (2003).
155 2.9 LandauLevelsandQuasiparticlesinMagneticField
Inthisnewpicture,wecannowusethesameargumentationfortheintegerquantumHall effectinSection2.9.3tounderstandtheHalleffectofthenewquasiparticles.Aquasipar- ticlewithcharge −e/3willhaveasetofLandaulevelswiththreetimestheenergyspacing and one-thirdthe number of states. These new Landaulevels willalso be separatedby mobilitygaps,justastheintegerLandaulevelswere.Wenolongerworryabouttheunder- lying electrons (or the lattice of atoms, for that matter) and only worryabout the new
13-charge quasiparticles. Thesame approachapplies forelectrons with l = 5or higher, whichcorrespondtocompositefermionscomposedofanelectronboundtofourorsome largerevennumber offlux quanta. Agreat number of experimentshaveconfirmed this picture.Furthermore,newquasiparticlesconsistingofboundstatesofcompositefermions canalsobeformed.
Again,itistemptingtothinkthatthesenewquasiparticlesarenot“real.”Buttheyare realinthesensethattheycarrychargeandhavewell-definedLandaulevels.Thefractional chargeofthesequasiparticles hasbeendramaticallydemonstratedbytheobservation of shotnoiseofthecompositefermions(DePicciotto etal. 1997;Samindayar etal. 1997).(As discussedinSection9.5,theamplitudeofshotnoisedependsonthechargeoftheparticles.) Inotherwords,thecompositefermionscarrychargein“lumps”thatgive“clicks”justas electronsdo.Thisaffectsourunderstandingofallparticles.Evenif weobserveclicksin detectionapparatusforthearrivalofparticles,itdoes notmeanthattheyareindivisible, fundamentalentities.AsdiscussedinSection2.1,allparticlesmayultimatelybebuiltout ofotherparticlesinsomeunderlyingfield.
Thisdiscussionhighlightsthetwodifferenttypesofquasiparticlepicturethatwehave beenworkingwith.Inonepicture,wedefinethegroundstateofasystemasthevacuum, anddefinetheexcitationsoutofthegroundstateasthequasiparticles.Thiswasthecase forfreeelectronsandholesinasemiconductor.Inasecondpicture,wedefinethevacuum asthestatewithallbandseitheremptyorfull.Thegroundstateofthesystemthenconsists ofanumberofadditionalquasiparticlescreatedintheemptyband.Thiswasthecasefor ametal –in Section 2.4 we treated the state of the system in whichthere is anempty conductionband,andanynumberoffullbandsatlowerenergy,asthevacuum,andtreated the free electrons in the conduction band as quasiparticles created in that vacuum. We could,alternatively,adoptthefirstpicture,and definethegroundstateofthemetal, with theelectronsalreadyintheconduction band,asthevacuum,and considerholescreated belowtheFermilevelandelectronsexcitedoutofthegroundstateasthequasiparticles.
WewillreturntothesetwopicturesofametalinSection5.5.1.
The quasiparticles in the fractional quantum Hall effect are defined in the second picture. We start with an empty band and create the composite quasiparticles in that band;theyarenotexcitationsoutofthegroundstate,butinsteadformthegroundstate.
The unifying concept in all cases is that we can take an enormous amount of infor- mation about the underlying system and bury it in the definition of the vacuum and the quasiparticles, and then have a simple system in which only the quasiparticles are relevant.
Exercise2.9.5 Provethatthewavefunction(2.9.38) isaneigenstateofthetotalangular momentumoperatorfor N particles.
References
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G.Baym, LecturesonQuantumMechanics (Benjamin-Cummins,1969).
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W.Ebeling,W.-D.Kraft,andD.Kremp, TheoryofBoundStatesandIonizationEquilibria inPlasmasandSolids (Akademie-Verlag,1976).
K.Huang, StatisticalMechanics (Wiley,1963).
J.K.Jain,“TheroleofanalogyinunravelingthefractionalquantumHalleffectmystery,”
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W.-D.Kraft,D.Kremp,W.Ebeling,andG.Rửpke, QuantumStatisticsofChargedParticle Systems (Plenum,1986).
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3 Classical Waves in Anisotropic Media
Sofar,wehavestudiedthepropertiesofelectronsinsolids.Incomputingtheseproperties, wehaveeffectivelyassumedthatthesolidisastatic,unchangingbackground.Inreality, however,theatomsinasolidconstantlymoveinresponsetosoundwavesandelectromag- neticwaves.Treatingtheatomsinthesolidasstaticisareasonableapproximationifthey donotmovefarfromtheirequilibriumpositions.Eventually,however,wemust concern ourselveswiththeinteractionsofelectronswiththesewaves.Beforewedothis,wemust firstwritedowntheproperdescriptionofthewavesthemselves.
Inelementaryphysics,we learnsimplewavetheory,inwhichallmediaareisotropic, andallwavesobeySnell’slaw.Mostsolidsare anisotropic,however;thatis,someoftheir propertiesarenotthesameinalldirections.Thisshouldnotbesurprising,becausecrystals havesymmetryaxesthatpointincertaindirections,andthereforetheoverlapoftheatomic orbitalswillbedifferent indifferent directions. Ingeneral,thereis noparticularreason toexpectthat materialpropertieswillbethesameinalldirections.The isotropicmodel workswellonlyforgases,fluids,andstronglydisorderedsolids.
Inanisotropicmedia,Snell’slawbreaksdownforalltypesofwaves.Forexample,aray enteringamediumperpendiculartothesurfacemayberefractedtomoveoffatanangle, incompleteviolationofSnell’slaw,asshowninFigure3.1.
Bothsoundandlight haveunusualpropertiesinsolids.Wewillexaminesoundwaves first, and then light waves. In each case, the symmetry properties of crystals greatly simplifytheanalysisofwavesinthesemedia.