Dynamics of a classical Hall system driven by a ti

arXiv:math-ph/0609039v113Sep2006DynamicsofaclassicalHallsystemdrivenbyatime-dependentAharonov–BohmfluxJ.Asch∗,P.ˇSˇtov´iˇcek†04.09.2006AbstractWestudythedynamicsofaclassicalparticlemovinginapuncturedplaneundertheinfluenceofastronghomogeneousmagneticfield,anelectricalbackground,anddrivenbyatime-dependentsingularfluxtubethroughthehole.Weexhibitastrikingclassical(de)localizationeffect:inthefarpastthetrajectoriesarespiralsaroundaboundcenter;theparticlemovesinwardtowardsthefluxtubeloosingkineticenergy.Afterhittingthepunctureitbecomes“conducting”:themotionisacycloidaroundacenterwhosedriftisoutgoing,orthogonaltotheelectricfield,diffusive,andwithoutenergyloss.PACSnumbers:45.50.PkParticleorbitsclassicalmechanics,45.50.-jDy-namicsandkinematicsofaparticleandasystemofparticles,73.43.-fQuantumHalleffects,73.50.GrChargecarriers:generation,recombina-tion,lifetime,trapping,meanfreepaths1IntroductionThemotivationtostudythedynamicsofthisclassicalsystemistosharpenourintuitiononitsquantumcounterpartwhichis,followingLaughlin’s[14]and∗CPT-CNRS,LuminyCase907,F-13288MarseilleCedex9,France.e-mail:[e-mail]†DepartmentofMathematics,FacultyofNuclearScience,CzechTechnicalUniversity,Tro-janova13,12000Prague,CzechRepublic1Halperin’s[12]proposals,widelyusedforanexplanationoftheIntegerQuantumHalleffect.Ofspecialinterestishowthetopologyinfluencesonthedynamics.InthemathematicalphysicsliteratureBellissardetal.[5]andAvron,Seiler,Simon[3],[4]usedanadiabaticlimitofthemodeltointroduceindices.Theindicesexplainthequantizationofchargetransportobservedintheexperiments[13].See[7,10,8,9,11]forrecentdevelopments.Wediscussedtheadiabaticsofthequantumsystemin[2],itsquantumandsemiclassicaldynamicswillbetreatedelsewhere.Thedynamicsoftheclassicalsystemwithoutmagneticfieldwerediscussedin[1].Westatethemodelandourmainresults:Consideraclassicalpointparticleofmassm0andchargee0movinginthepuncturedplaneR2\(0).SupposethatamagneticfluxlinewithtimevaryingstrengthΦpiercestheoriginandfurtherthepresenceofahomogeneousmagneticfieldofstrengthB0orthogonaltotheplaneandaninteriorelectricfieldwithsmoothboundedpotentialV.TheequationsofmotionsareHamiltonian.Forapoint(q,p)=((q1,q2),(p1,p2))inphasespaceP=R2\(0)×R2thetimedependentHamiltonianis:12m(p−eA(t,q))2+eV(t,q);A(t,q)=B2−Φ(t)2π|q|2q⊥whereq⊥:=(−q2,q1).WesupposethatΦ:R→RandV:R×R2→Raresmoothfunctions.Theelectricfieldis−∂tA−∂qV,theforceontheparticlewithvelocity˙q:e(˙q∧rot(A)−∂tA−∂qV)=−eB˙q⊥−∂tΦ2πq⊥|q|2+∂qVRemarkthatthepartoftheelectricfieldinducedbythefluxhascirculatione∂tΦ2πbutvanishingrotation,andislongrangewithan1/rsingularityattheorigin,wecallitthecircularparts.Vissmoothontheentireplanesothatthecirculationofthecorrespondingfieldiszero.Thisisthetopologyessentialforthedynamics.2Recallthatwhenonlytheconstantmagneticfieldispresent,theparticlefollowstheLandauorbits;thesearecirclesaroundafixedcenterwithfrequencyeBmwhosesquaredradiusisproportionaltotheenergy.OurresultforthecaseΦ∼t,Blarge,Vsuchthatthetorqueq∧∂qVissmallisqualitatively:–themotioninconfigurationspaceisapproximatelyrotationwithradiusproportionaltothesquarerootofthe(time-dependent)energyaroundadriftingcenter.–forlargeenoughnegativetimesthecenteristrappedbythefluxlineandtheenergyislinearlydecreasingwithtime,sotheparticleisspiralinginwards–fromthehittingtimeon(i.e.thetimewhentheLandauorbit“hits”thesingularity)thecenterstartstodriftawayfromthefluxline,theenergyremainsasymptoticallyconstantinthefuture.Thedriftisdiffusive.ThesituationisdescribedbyFig.1,showingatypicalorbitinq–space.RemarkthatthecorrespondinganalysisremainstrueifthesignofBischanged.Inthiscasewemaystateourobservationas:Hallconductingstatesareeventuallytrappedbythefluxlineandtrappedstatesareenergyconducting.Here“hallconducting”meansthatthecenterfollowsthelinesofthepotentialdiffusively.Weshalldiscussthecorrespondingquantumbehaviorelsewhere.Inthefirstsectionofthispaperwestatesomegeneralremarksonthemodelanddiscusstheproblemforfrozenvaluesoftheflux.Nextwedefineappropriateactionanglecoordinatesanduseanaveraging(adiabatic)methodtoapproximatethedynamicsnearthehittingtimebetweentheparticleandthefluxline.Inthelastsectionwediscusstheasymptoticbehaviorofthesolutionofthefullequationsofmotion.Letusremarkthatourmethodincludes(forthetwodimensionalcase)asimpleprooffortheguidingcenterapproximationwidelyusedinplasmaphysics.2DynamicsofthefrozensystemDenoteω=eBm,λ=1√eB.3Figure1:TypicaltrajectoryoftheHamiltonian12p−12q⊥−sq⊥q2+s∂qV2withVchosentobeV(x,y)=1/3(sinx+siny)4Weusethescaling(t,q,p)7→(ωt,q/λ,pλ)and“absorb”Vintothetimedepen-dentvectorpotential.Thescaledvariablesarecalled(s,q,p).TheHamiltonianunderconsiderationthenreadsH(s;p,q):=12(p−a(s;q))2;a(s;q):=12q⊥+aE(s;q)whereaE(s):R2\(0)→R2issmoothlytimedependentwithrot(aE)(s)=0.aE(s)andtheelectricfieldE(s):R2\(0)→R2aredefinedby:−∂saE(s):=E(s):=1ω∂tΦ2πsωq⊥|q|2−λ(∂qV)sω,λq(1)Wediscussfirstthesolutionoftheequationofmotionsforafrozentimeσ∈R.As∂saE(σ;q)=0,thesolutionofthefrozenequationsgeneratedbytheHamiltonianH(σ)goesalongthelinesoftheclassicalLandauproblem(whichmeans:thecaseΦ=0;V(q)=0)Forσ∈Rdefi