random network models and quantum phase transitions in two dimensions

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1、PhysicsReports417(2005)211–342www.elsevier.com/locate/physrepRandomnetworkmodelsandquantumphasetransitionsintwodimensionsB.Kramera,∗,1,T.Ohtsukib,S.KettemannaaI.InstitutfürPhysik,UniversitätHamburg,Jungiusstraße9,20355Hamburg,GermanybDepartmentofPhysics,So

2、phiaUniversity,Kioi-cho7-1,Chiyoda-ku,Tokyo,JapanAccepted18June2005editor:C.W.J.BeenakkerDedicatedtoKlausvonKlitzingontheoccasionofthe25thanniversaryoftheQuantumHallEffectAbstractAnoverviewoftherandomnetworkmodelinventedbyChalkerandCoddington,anditsgeneral

3、izations,isprovided.AfterashortintroductionintothephysicsoftheIntegerQuantumHallEffect,whichhistoricallyhasbeenthemotivationforintroducingthenetworkmodel,thepercolationmodelforelectronsinspatialdimen-sion2inastrongperpendicularmagneticfieldandaspatiallycorr

4、elatedrandompotentialisdescribed.Basedonthis,thenetworkmodelisestablished,usingtheconceptsofpercolatingprobabilityamplitudeandtunnel-ing.Itslocalizationpropertiesanditsbehavioratthecriticalpointarediscussedincludingashortsurveyonthestatisticsofenergylevels

5、andwavefunctionamplitudes.Magneto-transportisreviewedwithemphasisonsomenewresultsonconductancedistributions.GeneralizationsareperformedbyestablishingequivalentHamil-tonians.Inparticular,thesignificanceofmappingstotheDiracmodelandthetwo-dimensionalIsingmodel

6、isdiscussed.Adescriptionofrenormalizationgrouptreatmentsisgiven.Theclassificationoftwo-dimensionalrandomsystemsaccordingtotheirsymmetriesisoutlined.Thisprovidesaccesstothecompletesetofquan-tumphasetransitionslikethethermalHalltransitionandthespinquantumHall

7、transitionintwodimensions.∗Correspondingauthor.E-mailaddress:kramer@physnet.uni-hamburg.de(B.Kramer).1Permanentaddress:InternationalUniversityBremen,Campusring1,28759Bremen,Germany.0370-1573/$-seefrontmatter©2005PublishedbyElsevierB.V.doi:10.1016/j.physrep

8、.2005.07.001212B.Krameretal./PhysicsReports417(2005)211–342Thesupersymmetriceffectivefieldtheoryforthecriticalpropertiesofnetworkmodelsisformulated.Thenetworkmodelisextendedtohigherdimensionsincludingremarkson