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Ek=Λak=ΛbEk=Λak=ΛbTheQuantumSpinHallEffectandTopologicalBandTheoryTheQuantumSpinHallEffectandTopologicalBandTheoryI.Introduction-TopologicalbandtheoryII.TwoDimensions:QuantumSpinHallInsulator-Timereversalsymmetry&EdgeStates-Experiment:TransportinHgCdTequantumwellsIII.ThreeDimensions:TopologicalInsulator-TopologicalInsulator&SurfaceStates-Experiment:PhotoemissiononBixSb1-xIV.Superconductingproximityeffect-Majoranafermionboundstates-Aplatformfortopologicalquantumcomputing?ThankstoGeneMele,LiangFu,JeffreyTeo,TheInsulatingStateCovalentInsulatorCharacterizedbyenergygap:absenceoflowenergyelectronicexcitationsThevacuumAtomicInsulatore.g.solidArDiracVacuumEgap~10eVe.g.intrinsicsemiconductorEgap~1eV3p4sSiliconEgap=2mec2~106eVelectronpositron~holeTheIntegerQuantumHallState2DCyclotronMotion,LandauLevelsIQHEwithzeronetmagneticfield(HaldanePRL1988)GrapheneinaperiodicmagneticfieldB(r)gapcEω=EHallConductanceσxy=ne2/hBandstructureB(r)=0Zerogap,DiracpointB(r)≠0Energygapσxy=e2/hEgapk+−+−+−++−+−++−+−+−+TopologicalBandTheoryg=0g=121()()Integer2BZnduuiπ=⋅∇×∇=∫kkkkkThedistinctionbetweenaconventionalinsulatorandthequantumHallstateisatopologicalpropertyofthemanifoldofoccupiedstatesAnalogy:Genusofasurface:g=#holes|():()kΨBrillouinzoneatorusHilbertspaceInsulator:n=0IQHEstate:σxy=ne2/hTheTKNNinvariantcanonlychangeataquantumphasetransitionwheretheenergygapgoestozeroClassifiedbytheChern(orTKNN)topologicalinvariant(Thoulessetal,1982)EdgeStatesGaplessstatesmustexistattheinterfacebetweentopologicallydistinctphasesIQHEstaten=1EgapDomainwallboundstateψ0v()()xxyyzHiMxσσσ=−∂+∂+0(')'0()xyMxdxikyxeeψ−∫∼Vacuumn=0Edgestates~skippingorbitsLeadtoquantizedtransportn=1n=0ChiralFermionscanonlyexistatadomainwall0()vyEkk=Modelasbandinversiontransition–DiracEq.xyM0M0GaplessChiralFermionsEHaldaneModelkyEgapKK’QuantumSpinHallEffectinGrapheneTheintrinsicspinorbitinteractionleadstoasmall(~10mK-1K)energygapSimplestmodel:|Haldane|2(conservesSz)Haldane*Haldane0000HHHHH↑↓⎛⎞⎛⎞==⎜⎟⎜⎟⎝⎠⎝⎠EdgebandstructureTwoTerminalConductance:22eIVh=Theedgestatesareprotectedbytimereversalsymmetry•Kramers’degeneracyatk=π/aprotectsbandcrossing.•ElasticBackscatteringisforbidden.No1Dlocalization•ConservationofSzisNOTessentialKaneandMelePRL2005SpinFilterededgestatesJ↑J↓E↑↓↑↓↑↓TopologicalInsulator:ANewB=0Phase2DTimereversalinvariantbandstructureshaveaZ2topologicalinvariant,ν=0,1ν=0:ConventionalInsulatorν=1:TopologicalInsulatorKramersdegenerateattimereversalinvariantmomentak*=−k*+GEk*=0k*=π/aEk*=0k*=π/aEdgeStates1.Spinrotationsymmetry:Szconservedνisapropertyofbulkbandstructure.Easiesttocomputeifthereisextrasymmetry:,mod2nν↑↓=2.Inversion(P)Symmetry:determinedbyParityofoccupied2DBlochstatesatΓ1,2,3,4()()()()1ninininiPψξψξΓ=ΓΓΓ=±Γ4Γ3Γ1Γ2421(1)()niinυξ=−=Γ∏∏BulkBrillouinZonenn↑↓=−QuantumspinHalleffectIndependentspinChernintegersQuantumSpinHallEffectinHgTequantumwellsBernevig,HughesandZhang,Science‘06HgTeHgxCd1-xTeHgxCd1-xTedd6.3nm:Normalbandorderd6.3nm:InvertedbandorderConventionalInsulatorQuantumspinHallInsulatorwithtopologicaledgestatesΓ6~sΓ8~pkEΓ6~sΓ8~pkEEgap~10meVObservationofQSHinsulatorinHgTequantumWellsKonig,Wiedmann,Brune,Roth,Buhmann,Molenkamp,Qi,ZhangScience2007Measure2terminalconductancewhiletuningEFthroughthebulkenergygapI.d=5.5nm(normal)InsulatingingapII-IVd=7.3nm(inverted)conductingingapII.L=20µm(Lin)III.L=1µmW=1µmIV.L=1µmW=.5µm•Edgestateconductance2e2/hobservedindependentofWforshortsamples(LLin)•BulkbandinversionconfirmedbyLandaulevelevolutioninmagneticfield•MagneticfieldsuppressesedgetransmissionWLdThreeDimensionalTopologicalInsulatorsIn3Dthereare4Z2invariants:(ν0;ν1ν2ν3)characterizingthebulk.Thesedeterminehowsurfacestatesconnect.Fu,Kane&MelePRL07Moore&BalentsPRB07Roy,cond-mat06SurfaceBrillouinZoneΛ4Λ1Λ2Λ32DDiracPointEk=Λak=ΛbEk=Λak=Λbν0=1:StrongTopologicalInsulatorFermiarcenclosesoddnumberofDiracpointsTopologicalMetal•Berry’sphaseπaroundFermiarc•Robusttodisorder(antilocalization)•“violates”Fermiondoublingtheoremν0=0:WeakTopologicalInsulatorFermiarcenclosesevennumberofDiracpointsNormalMetal•Berry’sphase0,lessrobust.OREFBi1-xSbxPredictBi1-xSbxisastrongtopologicalinsulator:(1;111).0821(1)()niinυξ=−=Γ∏∏Inversionsymmetry⇒EFPureBismuthsemimetalAlloy:.09x.18semiconductorEgap~30meVPureAntimonysemimetalLsLaLsLaLsLaEFEFEgapTLTLTLEkExperimentsonBi1-xSbxMapE(kx,ky)for(111)surfacestatesbelowEFusingAngleResolvedPhotoemissionSpectroscopy•BulkDiracpointsatLprojecttoMinsurfaceBrillouinZone•Observe5surfacestatebandscrossingEFbetweenΓandMandKramersdegeneratesurfaceDiracpointatM.•Bi1-xSbxisaStrongTopologicalInsulatorD.Hsieh,D.Qian,L.Wray,Y.Xia,Y.S.Hor,R.J.CavaandM.Z.Hasan,Nature(08)inpressSuperconductingProximityEffectswavesuperconductorTopologicalinsulator2DinterfacestatewithenergygapandexotictopologicalorderResembles2Dspinlesspx+ipysuperconductorbutdoesnotviolatetimereversalsymmetry††*†(v)Hiψσµψψψψψ↑↓↓↑∆+=−∇∆−+iproximityinducedsuperconductivityDiracsurfacestates(nospindegeneracy)()ireθ∆=∆Majoranaboundstateatavortex:-boundstatesolutiontoBdGequationatexactlyzeroenergy-c0=c0†(electron=ho