Energetic and dynamic properties of a quantum part

arXiv:math-ph/0507035v115Jul2005Toappearin:AnnalesHenriPoincar´eENERGETICANDDYNAMICPROPERTIESOFAQUANTUMPARTICLEINASPATIALLYRANDOMMAGNETICFIELDWITHCONSTANTCORRELATIONSALONGONEDIRECTIONHAJOLESCHKE,SIMONEWARZEL,ANDALEXANDRAWEICHLEINABSTRACT.WeconsideranelectricallychargedparticleontheEuclideanplanesub-jectedtoaperpendicularmagneticfieldwhichdependsonlyononeofthetwoCartesianco-ordinates.Forsucha“unidirectionallyconstant”magneticfield(UMF),whichother-wisemayberandomornot,weprovecertainspectralandtransportpropertiesassoci-atedwiththecorrespondingone-particleSchr¨odingeroperator(withoutscalarpotential)byanalysingits“energy-bandstructure”.Inparticular,foranergodicrandomUMFweprovideconditionswhichensurethattheoperator’sentirespectrumisalmostsurelyab-solutelycontinuous.Thisimpliesthat,alongthedirectioninwhichtherandomUMFisconstant,thequantum-mechanicalmotionisalmostsurelyballistic,whileintheperpen-diculardirectionintheplaneonehasdynamicallocalisation.Theconditionsareverified,forexample,forGaussianandPoissonianrandomUMF’swithnon-zeromean-values.Theseresultsmaybeviewedas“randomanalogues”ofresultsfirstobtainedbyA.Iwat-suka[Publ.RIMS,KyotoUniv.21(1985)385]and(non-rigorously)byJ.E.M¨uller[Phys.Rev.Lett.68(1992)385].InmemoriamHeinzBAUER(31January1928–15August2002)formerProfessorofMathematicsattheUniversityofErlangen-N¨urnbergCONTENTS1.Introduction22.Schr¨odingeroperatorswithunidirectionallyconstantmagneticfields32.1.Energybandsandrelatedspectralproperties42.2.Energybandsandsometransportproperties93.Schr¨odingeroperatorswithrandomunidirectionallyconstantmagneticfields123.1.Non-randomnessoftheenergybands143.2.Moreontheenergybandsinthesign-definitecase153.3.Ontheabsenceofflatenergybandsinthenon-sign-definitecase163.4.Examples17AppendixA.OnthetopologicalsupportofcertainGaussianpathmeasures19Acknowledgements22References22Date:July,15,2005.12HAJOLESCHKE,SIMONEWARZEL,ANDALEXANDRAWEICHLEIN1.INTRODUCTIONThequantum-dynamicalbehaviourofelectricallychargedparticlesinaspatiallyrandommagneticfield(RMF)hasbecomeatopicofgrowinginterestoverthelastdecade.Mosttheoreticalinvestigationsofcorrespondingone-particlemodelstaketheirmotivationfromthephysicsof(quasi-)two-dimensionalsystems.Forexample,inconnectionwiththefractionalquantumHalleffect,transportpropertiesofinteractingelectronsonthe(infinitely-extended)EuclideanplaneR2subjectedtoanexternalrandomscalarpotentialandaperpendicular,stronghomogeneousmagneticfieldareoftendescribedby(non-interacting)effective,so-calledcompositefermionsinaRMF,whichishomogeneousonaverage.NearhalffillingofthelowestLandaulevel,thevaluesofthis(fictitious)RMFfluctuateateachpointx=(x1,x2)∈R2aboutamean-valuenearzero[24,70,47].Moreover,experimentalrealisationsofgasesofnon-interactingfermionsin(actual)RMF’sbyquasi-two-dimensionalsemiconductorheterostructureswithcertainrandomlybuilt-inmagnetshavebeenreported[20,63,44,3,57,9,58].Lastbutnotleast,thereisafundamentalinterestinthetheoryofone-particlemodelswithRMF’sintwodimensions.JustlikeinAnderson’sproblem[2]ofaquantumparticlesubjectedtoarandomscalarpotential(only),animportantquestioniswhetherall(generalised)energyeigenstatesarespatiallylocalisedorwhethersomeofthemaredelocalised.Untilrecently,intheRMF-casetheanswertothequestionhasremainedcontroversialwithinperturbative,quasi-classical,field-theoreticalandnumericalstudies[4,40,32,59,72,6,17,19,51,31,60,71,65,48,16,33].Itisthereforedesirabletoestablishexactlocalisation/delocalisationresultsfortheRMF-caseashasbeendoneforrandomscalarpotentials[10,50,64](seealso[41]).FortheRMF-case(withoutarandomscalarpotential)weareawareofonlyonerigorouswork[35]devotedtothelocalisation/delocalisationproblem.ThereinKlopp,Nakamura,NakanoandNomuraoutlineaproofoftheexistenceoflocalisedstatesatlowenergiesinacertainmodelforaparticleonthe(unit-)squarelatticeZ2insteadofthetwo-dimensionalcontinuumR2.Inthepresentpaperweprovefirstexactlocalisation/delocalisationresultsforasimplifiedmodelforaparticleonthecontinuumR2.ThesimplificationarisesfromtheassumptionthatthefluctuationsoftheRMFonR2areanisotropicallylong-rangedcor-relatedinthesensethatweconsiderthelimitingcaseofaninfinitecorrelationlengthalongonedirectionandtakethecorrelationlengthtobefinitebutstrictlypositivealongtheperpendiculardirectionintheplane.Inotherwords,theRMFisassumedtobeinde-pendentofoneofthetwoCartesiancoordinates,whichwechoosetobethesecondone,x2.TheremainingdependenceoftheRMF-valuesonthefirstcoordinatex1wesupposetobegovernedbytherealisationsofanergodicreal-valuedrandomprocesswiththereallineRasitsparameterset.Fortheprecisedescriptionofsucharandomunidirection-allyconstantmagneticfield(RUMF)seeDefinition3.1below.Toourknowledge,thefirstrigorousworkexplicitlydealingwithamodelinvolvingarandomUMF(withzeromean-value)isoneofUeki[67].ModelsforasingleparticleontheplaneR2subjectedtoanon-randomunidirec-tionallyconstantmagneticfield(UMF)havebeentheobjectofvariousstudiesinthemathematics[28,13,45,42]andphysics[46,49,37,56,61,39]literature.ThesemodelsENERGETICANDDYNAMICPROPERTIES3illustrateunadulteratedlythatinhomogeneousmagneticfieldshave