Chern-Simons Theory of Fractional QuantumHall EectKalinVetsigianMay7,2001AbstractTheFractional QuantumHall eectisreviewedfromtheperspectiveof Chern-Simons eld theory. The interacting 2D electron gas in magneticeld problem is exactly mapped to a bosonic problem in which the bosonscoupletoanewgaugeeldinadditiontotheelectromagneticeld. ItisshownthatmeaneldanalysisinthenewformulationissucienttoexplainallbasicfeaturesofthefractionalquantumHalleect.1 IntroductionThe theory of Fractional Quantum Hall Eect (FQHE) provides us with a strik-ingexampleofastronglyinteractingsystemwhichcanbewellunderstoodintermsof theweakinteractionsof nontrivial eectiveobjects. Whatismoststriking about FQHE is that due to their fractional charge the eective objects,unlike atoms and molecules, cannot be thought of as a simple collection of moreelementary particles.The FQHE is basically the problem of interacting electrons in 2D in the pres-ence of a strong perpendicular magnetic eld. The ground state wave functionis completely dierent from the non-interacting one and the problem cannot beunderstood by means of perturbation theory starting from the non-interactingcase. The many body ground state wave function was guessed by Laughlin withthe help of the variational principle. Once this highly non-trivial step was madethe fractional Hall eect theory developed quickly and many dierent ways oflooking at the problem were invented. In this paper we will review a work donebyZhang, HanssonandKivelson[1][2] whomappedtheinteractingelectronproblem to one of interacting bosons coupled to an additional gauge eld. Theadvantage of the new formulation is that near lling factors of the form 1/(2k+1)the essential features of FQHE can be derived by a straightforward mean eldanalysis. This formulation enables us to compute quantities in a systematicallyimprovable way using the machinery of perturbation theory.Chern-Simons Landau-Ginzburg (CSLG) approach to the FQHE makes useof the fact that in 2 dimensions an electron can be treated as a boson to which anodd number of magnetic ux quanta are attached. For lling factors = 1/(2k+11) the ground state is just a homogeneous bosonic eld. The topologically trivialuctuations around the uniform state are gaped as can be easily seen from theclassical equations of motions for a charged liquid. The lowest lying excitationsaretopological vorticesintheuniformstate. Thesevorticescarryfractionalchargee/(2k + 1). Theresistivedissipationifpresentcomesfromthemotionof the vortices. Under conditions in which FQHE eect is observed the vorticesare pinned down by the eects of disorder just like in type II superconductors.When the vortices are free to move they behave as quasi particles with fractionalchargeandfractional statistics. Thesequasi particlescaninturncondensatecreating a hierarchy of FQHE states. This hierarchy can explain FQHE at llingfractionsdierentfrom=1/(2k + 1). If theaverageelectrondensitydoesnot correspond exactly to a lling factor of= 1/(2k + 1) the extra charge isaccommodated by creating localized vortices in the otherwise uniform Bose eld.ThisisanalogoustothewayinwhichtypeIIsuperconductorsaccommodateextra magnetic eld.2 ExperimentalObservationsAn eectively 2D electron system is created at the interface of a semiconductorandaninsulatorortwosemiconductors(oneofthemactingasaninsulator).The electrons are trapped in a quantum well in direction perpendicular to thesurface formed by the insulator(acting as a high barrier) and an applied electriceld perpendicular to the interface. The quantum well is narrow enough so thatthe z dependence of the wave function is quantized and at low temperatures thez-dependence of the wave function is xed to the lowest level.Acurrentisappliedtothe2Dsystem, andtheresultingHall voltageinthe perpendicular direction is measured. It is observed that for certain samplesand certain applied perpendicular magnetic elds the transverse conductivity isxy=fe2/h withfa rational fraction and at the same time the longitudinalconductivityxx=0- bothwithaveryhighaccuracy. This is adeningcharacteristic of the Hall eect.Let denotetheelectrondensityandBtheappliedmagneticeldinz-direction. The plateaus are formed around lling factors hc/(eB) whichare rational fractions. The plateaus are most prominent at fractions of the type1/(2k + 1), k = 0, 1, 2, ...The other fundamental aspect of the fractional quantum Hall eect is the ex-istence of fractionally charged quasi particles. The charge of the quasi particlesisjusttheelectronchargetimesthefractionfforthecorrespondingplateau.The charge of the quasi particles can be measure directly using a device calledquantum antidot electrometer[3]. The most notable result is that the fractionalchargeisthesamewithinagivenplateau,ex. forllingfactorsclosebutnotequal to 1/3 the quasi particles still have charge e/3.23 Chern-SimonstheoryofFQHE3.1 MappingtoBosonicProblemThemicroscopicHamiltonianforacollectionof electronsinexternal electro-magnetic eld (A0, A) isH =

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