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1、ANNALSOFPHYSICS160,343-354(1985)TopologicalInvariantandtheQuantizationoftheHallConductanceMAHITOKOHMOTO*DepartmentofPhysicsandtheMaterialsResearchLaboratory,UniversityofIllinoisatUrbana-Champaign,Urbana,Illinois61801ReceivedMarch27,1984Thetopologicalaspectsofwavefunctionsforel
2、ectronsinatwodimensionalperiodicpoten-tialwithamagneticfieldarediscussed.SpecialattentionispaidtothelinearresponseformulafortheHallconductancecxY.ItisshownthatthequantizedvalueofgxYisrelatedtothenum-berofzerosofwavefunctionsinthemagneticBrillouinzone.Aphaseofwavefunctionscanno
3、tbedeterminedinauniqueandsmoothwayovertheentiremagneticBrillouinzoneunlessthemagneticsubbandcarriesnoHallcurrent.,c:1985AcademicPress.Inc.1.INTRODUCTIONThediscoveryofthequantizationoftheHallconductancebyvonKlitzing,Dorda,andPepper[l]isoneofthemostimportantresultsincondensedmat
4、terphysicsinrecentyears.Sincetherehadbeennopredictionofsuchaprecisequan-tization,thesubjecthasbeenunderintensivetheoreticalinvestigation[2].TheidealtwodimensionalelectronsystemhasintegralvaluesoftheHallconductanceinunitofe2/hiftheFermienergyliesinagapbetweenLandaulevels.Theexp
5、erimentalresultisastonishingbecause,inarealsystem,wewouldexpectcorrec-tionsofvarioussorts,dueto,forexample,electronelectroninteractions,impurities,quasi-two-dimensionality,substratepotentials,finitesizeofsamples,etc.Theeffectsofperiodicpotentialshavebeendiscussedbyseveralautho
6、rs[3-61.SpectrumofelectronsinacrystalwithamagneticfieldcandisplayanamazingcomplexityincludingvariouskindsofscalingandaCantorsetstructure[7-91.However,itwasshownthattheHallconductanceisstillanintegralmultipleofe2/haslongastheFermienergyliesinagap.Thispaperdiscussestopologicalas
7、pectsofatwo-dimensionalperiodicsystemsinamagneticfield.TheHallconductanceisshowntoberepresentedbyatopologicalinvariantwhichisnaturallyaninteger.ThisisasequeltoRef.[3]inaspiritsimilartoLaughlin’sarguments[lo]thatanexactquantizationmustbeaconsequenceofageneralprinciplewhichisdet
8、erminedbythegeometricalnatureoftheproblem.*Currentaddress:Dep
