Topological insulators and superconductors

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TopologicalinsulatorsandsuperconductorsXiao-LiangQi1,2andShou-ChengZhang21MicrosoftResearch,StationQ,ElingsHall,UniversityofCalifornia,SantaBarbara,CA93106,USA2DepartmentofPhysics,StanfordUniversity,Stanford,CA94305Topologicalinsulatorsarenewstatesofquantummatterwhichcannotbeadiabaticallyconnectedtoconventionalinsulatorsandsemiconductors.Theyarecharacterizedbyafullinsulatinggapinthebulkandgaplessedgeorsurfacestateswhichareprotectedbytime-reversalsymmetry.Thesetopologi-calmaterialshavebeentheoreticallypredictedandexperimentallyobservedinavarietyofsystems,includingHgTequantumwells,BiSballoys,andBi2Te3andBi2Se3crystals.Wereviewtheoreti-calmodels,materialspropertiesandexperimentalresultsontwo-dimensionalandthree-dimensionaltopologicalinsulators,anddiscussboththetopologicalbandtheoryandthetopologicalfieldtheory.TopologicalsuperconductorshaveafullpairinggapinthebulkandgaplesssurfacestatesconsistingofMajoranafermions.Wereviewthetheoryoftopologicalsuperconductorsincloseanalogytothetheoryoftopologicalinsulators.PACSnumbers:73.20.-r,73.43.-f,85.75.-d,74.90.+nCONTENTSA.Topologicalfieldtheory351.Chern-Simonsinsulatorin2+1dimensions35I.Introduction12.Chern-Simonsinsulatorin4+1dimensions353.Dimensionalreductiontothethree-dimensionalZ2II.Two-DimensionalTopologicalInsulators4topologicalinsulator36A.Effectivemodelofthetwo-dimensionaltime-reversalinvariant4.Furtherdimensionalreductiontothetwo-dimensionalZ2topologicalinsulatorinHgTe/CdTequantumwells5topologicalinsulator39B.Explicitsolutionofthehelicaledgestates75.GeneralphasediagramoftopologicalMottinsulatorandC.Physicalpropertiesofthehelicaledgestates8topologicalAndersoninsulator391.Topologicalprotectionofthehelicaledgestates8B.Topologicalbandtheory402.Interactionsandquencheddisorder10C.Reductionfromtopologicalfieldtheorytotopologicalband3.Helicaledgestatesandtheholographicprinciple10theory424.Transporttheoryofthehelicaledgestates11V.TopologicalSuperconductorsandSuperfluids42D.Topologicalexcitations12A.Effectivemodelsoftime-reversalinvariantsuperconductors431.Fractionalchargeontheedge12B.Topologicalinvariants452.Spin-chargeseparationinthebulk13C.Majoranazeromodesintopologicalsuperconductors46E.QuantumanomalousHallinsulator141.Majoranazeromodesinp+ipsuperconductors46F.Experimentalresults152.Majoranafermionsinsurfacestatesofthetopological1.Quantumwellgrowthandthebandinversiontransition15insulator472.LongitudinalconductanceinthequantumspinHallstate173.MajoranafermionsinsemiconductorswithRashbaspin-orbit3.MagnetoconductanceinthequantumspinHallstate18coupling484.Nonlocalconductance194.MajoranafermionsinquantumHallandquantumanomalousHallinsulators48III.Three-DimensionalTopologicalInsulators195.DetectionofMajoranafermions49A.Effectivemodelofthethree-dimensionaltopologicalinsulator20B.SurfacestateswithasingleDiraccone22VI.Outlook50arXiv:1008.2026v1[cond-mat.mes-hall]12Aug2010C.Crossoverfromthreedimensionstotwodimensions23D.Electromagneticproperties24ACKNOWLEDGMENTS501.HalfquantumHalleffectonthesurface252.Topologicalmagnetoelectriceffect26References513.Imagemagneticmonopoleeffect274.TopologicalKerrandFaradayrotation285.Relatedeffects29I.INTRODUCTIONE.Experimentalresults301.Materialgrowth302.Angle-resolvedphotoemissionspectroscopy30EversincetheGreeksinventedtheconceptoftheatom,fun-3.Scanningtunnelingmicroscopy31damentalsciencehasfocusedonfindingeversmallerbuild-4.Transport32ingblocksofmatter.Inthe19thcentury,thediscoveryofF.Othertopologicalinsulatormaterials34elementsdefinedthegoldenageofchemistry.ThroughoutIV.GeneralTheoryofTopologicalInsulators34mostofthe20thcentury,fundamentalsciencewasdominated 2bythesearchforelementaryparticles.Incondensedmatterprocess.Theoperationofsmoothdeformationgroupsshapesphysics,therearenonewbuildingblocksofmattertobedis-intotopologicalequivalenceclasses.Inphysics,onecancon-covered:oneisdealingwiththesameatomsandelectronssidergeneralHamiltoniansofmany-particlesystemswithanasthosediscoveredcenturiesago.Rather,oneisinterestedenergygapseparatingthegroundstatefromtheexcitedstates.inhowthesebasicbuildingblocksareputtogethertoformInthiscase,onecandefineasmoothdeformationasachangenewstatesofmatter.ElectronsandatomsinthequantumintheHamiltonianwhichdoesnotclosethebulkgap.Thisworldcanformmanydifferentstatesofmatter:forexample,topologicalconceptcanbeappliedtobothinsulatorsandsu-theycanformcrystallinesolids,magnetsandsuperconduc-perconductorswithafullenergygap,whicharethefocusoftors.Thegreatesttriumphofcondensedmatterphysicsinthethisreviewarticle.Itcannotbeappliedtogaplessstatessuchlastcenturyistheclassificationofthesequantumstatesbyasmetals,dopedsemiconductors,ornodalsuperconductors.theprincipleofspontaneoussymmetrybreaking(Anderson,Accordingtothisgeneraldefinition,ifweputincontacttwo1997).Forexample,acrystallinesolidbreakstranslationsym-quantumstatesbelongingtothesametopologicalclass,thein-metry,eventhoughtheinteractionamongitsatomicbuild-terfacebetweenthemdoesnotneedtosupportgaplessstates.ingblocksistranslationallyinvariant.Amagnetbreaksrota-Ontheotherhand,ifweputincontacttwoquantumstatesbe-tionsymmetry,eventhoughthefundamentalinteractionsarelongingtodifferenttopologicalclasses,orputatopologicallyisotropic.Asuperconductorbreaksthemoresubtlegaugenontrivialstateincontactwiththevacuum,theinterfacemustsymmetry,leadingtonovelphenomenasuchasfluxquantiza-supportgaplessstates.tionandJosephsoneffects.ThepatternofsymmetrybreakingFromthesesimplearguments,weimmediatelyseethattheleadstoauniqueorderparameter,whichassumesanonvan-abstractconceptoftopologicalclassificationcanbeappliedtoishingexpectationvalueonlyintheorderedstate,andagen-condensedmattersystemwithanenergygap,wheretheno-eraleffectivefieldtheorycanbeformulatedbasedontheordertionofasmoothdeformationcanbedefined(Zhang,2008).parameter.Theeffectivefieldtheory,generallycalledLandau-Furtherprogresscanbemadethroughtheconceptsoftopolog-Ginzburgtheory(LandauandLifshitz,1980),isdeterminedicalorderparameterandtopologicalfieldtheory(TFT),whichbygeneralpropertiessuchasdimensionalityandsymmetryarepowerfultoolsdescribingtopologicalstatesofquantumoftheorderparameter,andgivesauniversaldescriptionofmatter.Mathematicianshaveexpressedtheintuitiveconceptquantumstatesofmatter.ofgenusintermsofanintegral,calledtopologicalinvari-In1980,anewquantumstatewasdiscoveredwhichdoesant,overthelocalcurvatureofthesurface(Nakahara,1990).notfitintothissimpleparadigm(vonKlitzingetal.,1980).InWhereastheintegranddependsondetailsofthesurfacege-thequantumHall(QH)state,thebulkofthetwo-dimensionalometry,thevalueoftheintegralisindependentofsuchdetails(2D)sampleisinsulating,andtheelectriccurrentiscarriedanddependsonlyontheglobaltopology.Inphysics,topo-onlyalongtheedgeofthesample.Theflowofthisunidi-logicallyquantizedphysicalquantitiescanbesimilarlyex-rectionalcurrentavoidsdissipationandgivesrisetoaquan-pressedasinvariantintegralsoverthefrequency-momentumtizedHalleffect.TheQHstateprovidedthefirstexampleofspace(Thouless,1998;Thoulessetal.,1982).Suchquanti-aquantumstatewhichistopologicallydistinctfromallstatestiescanserveasatopologicalorderparameterwhichuniquelyofmatterknownbefore.TheprecisequantizationoftheHalldeterminesthenatureofthequantumstate.Furthermore,theconductanceisexplainedbythefactthatitisatopologicalin-long-wavelengthandlow-energyphysicscanbecompletelyvariant,whichcanonlytakeintegervaluesinunitsofe2/h,in-describedbyaTFT,leadingtopowerfulpredictionsofexperi-dependentofmaterialdetails(Laughlin,1981;Thoulessetal.,mentallymeasurabletopologicaleffects(Zhang,1992).Topo-1982).Mathematicianshaveintroducedtheconceptoftopo-logicalorderparametersandTFTsfortopologicalquantumlogicalinvariancetoclassifydifferentgeometricalobjectsintostatesplaytheroleofconventionalsymmetry-breakingorderbroadclasses.Forexample,2Dsurfacesareclassifiedbytheparametersandeffectivefieldtheoriesforbroken-symmetrynumberofholesinthem,orgenus.Thesurfaceofaperfectstates.sphereistopologicallyequivalenttothesurfaceofanellip-TheQHstatesbelongtoatopologicalclasswhichex-soid,sincethesetwosurfacescanbesmoothlydeformedintoplicitlybreakstime-reversal(TR)symmetry,forexample,eachotherwithoutcreatinganyholes.Similarly,acoffeecupbythepresenceofamagneticfield.Inrecentyears,istopologicallyequivalenttoadonut,sincebothofthemcon-anewtopologicalclassofmaterialshasbeentheoreti-tainasinglehole.Inmathematics,topologicalclassificationcallypredictedandexperimentallyobserved(Bernevigetal.,discardssmalldetailsandfocusesonthefundamentaldistinc-2006;Chenetal.,2009;Fuetal.,2007;Hsiehetal.,2008;tionofshapes.Inphysics,preciselyquantizedphysicalquan-K¨onigetal.,2007;Xiaetal.,2009;Zhangetal.,2009).titiessuchastheHallconductancealsohaveatopologicalori-Thesenewquantumstatesbelongtoaclasswhichisinvari-gin,andremainunchangedbysmallchangesinthesample.antunderTR,andwherespin-orbitcoupling(SOC)playsItisobviousthatthelinkbetweenphysicsandtopologyanessentialrole.Someimportantconceptsweredevel-shouldbemoregeneralthanthespecificcaseofQHstates.opedinearlierworks(BernevigandZhang,2006;Haldane,Thekeyconceptisthatofa“smoothdeformation”.Inmath-1988;KaneandMele,2005;Murakamietal.,2003,2004;ematics,oneconsiderssmoothdeformationsofshapeswith-Sinovaetal.,2004;ZhangandHu,2001),culminatingintheouttheviolentactionofcreatingaholeinthedeformationconstructionofthetopologicalbandtheory(TBT)andthe 3TFTof2Dand3Dtopologicalinsulators(FuandKane,2007;Fuetal.,2007;KaneandMele,2005;MooreandBalents,2007;Qietal.,2008;Roy,2009).AllTRinvariantinsula-torsinnature(withoutgroundstatedegeneracy)fallintotwodistinctclasses,classifiedbyaZ2topologicalorderparame-ter.Thetopologicallynontrivialstatehasafullinsulatinggapinthebulk,buthasgaplessedgeorsurfacestatesconsistingofanoddnumberofDiracfermions.Thetopologicalprop-ertymanifestsitselfmoredramaticallywhenTRsymmetryispreservedinthebulkbutbrokenonthesurface,inwhichcasethematerialisfullyinsulatingbothinsidethebulkandonthesurface.Inthiscase,Maxwell'slawsofelectrodynam-icsaredramaticallyalteredbyatopologicaltermwithapre-ciselyquantizedcoefficient,similartothecaseoftheQHef-FIG.1AnalogybetweenQHandQSHeffects:(a)Aspinless1Dfect.The2Dtopologicalinsulator,synonymouslycalledthesystemhasbothforwardandbackwardmovers.Thesetwobasicde-quantumspinHall(QSH)insulator,wasfirsttheoreticallypre-greesoffreedomarespatiallyseparatedinaQHbar,asexpressedbydictedin2006(Bernevigetal.,2006)andexperimentallyob-thesymbolicequation“2=1+1”.Theupperedgesupportsonlyaserved(K¨onigetal.,2007;Rothetal.,2009)inHgTe/CdTeforwardmoverandtheloweredgesupportsonlyabackwardmover.Thestatesarerobustandgoaroundanimpuritywithoutscattering.quantumwells(QW).Atopologicallytrivialinsulatorstate(b)Aspinful1Dsystemhasfourbasicdegreesoffreedom,whichisrealizedwhenthethicknessoftheQWislessthanacrit-arespatiallyseparatedinaQSHbar.Theupperedgesupportsafor-icalvalue,andthetopologicallynontrivialstateisobtainedwardmoverwithspinupandabackwardmoverwithspindown,andwhenthatthicknessexceedsthecriticalvalue.Inthetopolog-converselyfortheloweredge.Thatspatialseparationisexpressedicallynontrivialstate,thereisapairofedgestateswithop-bythesymbolicequation“4=2+2”.AdaptedfromQiandZhang,positespinspropagatinginoppositedirections.Four-terminal2010.measurements(K¨onigetal.,2007)showthatthelongitudinalconductanceintheQSHregimeisquantizedto2e2/h,inde-pendentlyofthewidthofthesample.Subsequentnonlocallogicalstateofquantummatter.Itisremarkablethatsuchtransportmeasurements(Rothetal.,2009)confirmtheedgetopologicaleffectscanberealizedincommonmaterials,pre-statetransportaspredictedbytheory.Thefirstdiscoveryofviouslyusedforinfrareddetectionorthermoelectricapplica-theQSHtopologicalinsulatorinHgTewasrankedbySci-tions,withoutrequiringextremeconditionssuchashighmag-enceMagazineasoneofthetoptenbreakthroughsamongneticfieldsorlowtemperatures.Thediscoveryoftopologicalallsciencesinyear2007,andthesubjectquicklybecameinsulatorshasundoubtedlyhadadramaticimpactonthefieldmainstreamincondensedmatterphysics(Day,2008).Theofcondensedmatterphysics.3DtopologicalinsulatorwaspredictedintheBi1−xSbxalloyAfterbrieflyreviewingthehistoryofthetheoreticalpre-withinacertainrangeofcompositionsx(FuandKane,2007),dictionandtheexperimentalobservationofthetopologicalandangle-resolvedphotoemissionspectroscopy(ARPES)materialsinnature,wenowturntothehistoryofthecon-measurementssoonobservedanoddnumberoftopolog-ceptualdevelopments,andretracetheintertwinedpathstakenicallynontrivialsurfacestates(Hsiehetal.,2008).Sim-bytheorists.Animportantstepwastakenin1988byHal-plerversionsofthe3Dtopologicalinsulatorweretheoret-dane(Haldane,1988),whoborrowedtheconceptoftheparityicallypredictedinBi2Te3,Sb2Te3(Zhangetal.,2009)andanomaly(Redlich,1984;Semenoff,1984)inquantumelectro-Bi2Se3(Xiaetal.,2009;Zhangetal.,2009)compoundswithdynamicstoconstructatheoreticalmodeloftheQHstateonalargebulkgapandagaplesssurfacestateconsistingofathe2Dhoneycomblattice.ThismodeldoesnotrequireansingleDiraccone.ARPESexperimentsindeedobservedtheexternalmagneticfieldnortheassociatedorbitalquantizationlineardispersionrelationofthesesurfacestates(Chenetal.,andLandaulevels(LLs).However,itisinthesametopo-2009;Xiaetal.,2009).Thesepioneeringtheoreticalandex-logicalclassastheordinaryQHstates,andrequiresbothtwoperimentalworksopeneduptheexcitingfieldoftopolog-dimensionalityandthebreakingoftheTRsymmetry.Thereicalinsulators,andthefieldisnowexpandingatarapidwasamisconceptionatthetimethattopologicalquantumpace(HasanandKane,2010;Kane,2008;K¨onigetal.,2008;statescouldonlyexistundertheseconditions.Anotherim-Moore,2009;QiandZhang,2010;Zhang,2008).Beyondportantstepwastheconstructionin1989ofaTFToftheQHthetopologicalmaterialsmentionedabove,morethanfiftyeffectbasedontheChern-Simons(CS)term(Zhang,1992).newcompoundshavebeenpredictedtobetopologicalin-Thistheorycapturesthemostimportanttopologicalaspectssulators(Chadovetal.,2010;Franz,2010;Linetal.,2010;oftheQHeffectinasingleandunifiedeffectivefieldthe-Yanetal.,2010),andtwoofthemhavebeenexperimentallyory.Atthispoint,thepathtowardsgeneralizingtheQHobservedrecently(Chenetal.,2010;Satoetal.,2010).Thisstatesbecameclear:sincetheCStermcanexistinallevencollectivebodyofworkestablishesbeyondanyreasonablespatialdimensions,thetopologicalphysicsoftheQHstatesdoubttheubiquitousexistenceinnatureofthisnewtopo-canbegeneralizedtosuchdimensions.However,itwasun- 4clearatthetimewhatkindofmicroscopicinteractionscouldTRinvariant2Dinsulators.Inaddition,theydevisedapre-beresponsibleforthesetopologicalstates.In2001,ZhangcisealgorithmforthecomputationofaZ2topologicalinvari-andHu(ZhangandHu,2001)explicitlyconstructedamicro-antwithinTBT.TBTwassoonextendedto3DbyFu,KanescopicmodelforthegeneralizationoftheQHstatein4D.andMele,MooreandBalentsandRoy(FuandKane,2007;AcrucialingredientofthismodelisitsinvarianceunderTRFuetal.,2007;MooreandBalents,2007;Roy,2009),wheresymmetry,insharpcontrasttotheQHstatein2Dwhichex-sixteentopologicallydistinctstatesarepossible.MostoftheseplicitlybreaksTRsymmetry.Thisfactcanalsobeseendi-statescanbeviewedasstacked2DQSHinsulatorplanes,butrectlyfromtheCSeffectiveactionin4+1spacetimedimen-oneofthem,thestrongtopologicalinsulator,isgenuinely3D.sions,whichisinvariantunderTRsymmetry.Withthisgener-ThetopologicalclassificationaccordingtoTBTisonlyvalidalizationoftheQHstate,twobasicobstacles,thebreakingoffornoninteractingsystems,anditwasnotclearatthetimeTRsymmetryandtherestrictionto2D,wereremoved.Partlywhetherthesestatesarestableundermoregeneraltopologicalbecauseofthemathematicalcomplexityinvolvedinthiswork,deformationsincludinginteractions.Qi,HughesandZhangitwasnotappreciatedbythegeneralcommunityatthetime—introducedtheTFToftopologicalinsulators(Qietal.,2008),butisclearnow—thatthisstateistherootstatefromwhichanddemonstratedthatthesestatesareindeedgenerallysta-allTRinvarianttopologicalinsulatorsin3Dand2Darede-bleinthepresenceofinteractions.Furthermore,atopolog-rived(Qietal.,2008).TRinvarianttopologicalinsulatorscanicallyinvarianttopologicalorderparametercanbedefinedbeclassifiedintheformofafamilytree,wherethe4DstatewithintheTFTasaexperimentallymeasurable,quantizedisthe“grandfather”stateandbegetsexactlytwogenerationstopologicalmagnetoelectriceffect.ThestandardMaxwell'sofdescendants,the3Dand2Dtopologicalinsulators,bytheequationsaremodifiedbythetopologicalterms,leadingtoprocedureofdimensionalreduction(Kitaev,2009;Qietal.,theaxionelectrodynamicsofthetopologicalinsulators.This2008;Ryuetal.,2010;Schnyderetal.,2008).workalsoshowedthatthe2Dand3DtopologicalinsulatorsMotivatedbytheconstructionofaTRinvarianttopolog-aredescendantsofthe4Dtopologicalinsulatorstatediscov-icalstate,theoristsstartedtolookforaphysicalrealizationeredin2001(ZhangandHu,2001),andmotivatedthisseriesofthisnewtopologicalclass,anddiscoveredtheintrinsicspinofrecentdevelopments.Atthispoint,thetwodifferentpathsHalleffect(Murakamietal.,2003,2004;Sinovaetal.,2004).basedontheTBTandTFTconverged,andanunifiedtheoret-Murakamietal.(Murakamietal.,2003)statetheirmotivationicalframeworkemerged.clearlyintheintroduction:“Recently,theQHeffecthasbeenThereareanumberofexcellentreviewsonthissub-generalizedtofourspatialdimensions[...].TheQHresponseject(HasanandKane,2010;K¨onigetal.,2008;Moore,2009;inthatsystemisphysicallyrealizedthroughtheSOCinaTRQiandZhang,2010).Thisarticleattemptstogiveasimplesymmetricsystem”.Soonafter,itwasrealizedin2004thatthepedagogicalintroductiontothesubjectandreviewsthecur-twokeyideas,TRsymmetryandSOC,canalsobeappliedtorentstatusofthefield.InSec.IIandSec.III,wereviewtheinsulatorsaswell,leadingtotheconceptofspinHallinsula-standardmodels,materialsandexperimentsforthe2Dandthetor(Murakamietal.,2004).ThespinHalleffectininsulators3Dtopologicalinsulators.Thesetwosectionscanbeunder-isdissipationless,similarlytotheQHeffect.Theconceptofstoodwithoutanypriorknowledgeoftopology.InSec.IV,spinHallinsulatormotivatedKaneandMelein2005toinves-wereviewthegeneraltheoryoftopologicalinsulators,pre-tigatetheQSHeffectingraphene(KaneandMele,2005),asentingboththeTFTandtheTBT.InSec.V,wediscussanmaterialfirstdiscoveredexperimentallythatsameyear.Work-importantgeneralizationoftopologicalinsulators–topologicalingindependently,BernevigandZhangstudiedtheQSHef-superconductors.fectinstrainedsemiconductors,whereSOCgeneratesLLswithoutthebreakingofTRsymmetry(BernevigandZhang,2006).UnfortunatelytheenergygapingraphenecausedbyII.TWO-DIMENSIONALTOPOLOGICALINSULATORStheintrinsicSOCisinsignificantlysmall(Minetal.,2006;Yaoetal.,2007).EventhoughneithermodelshavebeenTheQSHstate,orthe2Dtopologicalinsulatorwasfirstdis-experimentallyrealized,theyplayedimportantrolesforthecoveredintheHgTe/CdTequantumwells.Bernevig,Hughesconceptualdevelopments.In2006,Bernevig,HughesandandZhang(Bernevigetal.,2006)initiatedthesearchfortheZhang(Bernevigetal.,2006)successfullypredictedthefirstQSHstateinsemiconductorswithan“inverted”electronictopologicalinsulatortoberealizedinHgTe/CdTeQWs.gap,andpredictedaquantumphasetransitioninHgTe/CdTeTheQSHstatein2DcanberoughlyunderstoodastwoquantumwellsasafunctionofthethicknessdQWofthequan-copiesoftheQHstate,wherestateswithoppositespintumwell.Thequantumwellsystemispredictedtobeacon-counter-propagateattheedge.AnaturalquestionarisesasventionalinsulatorfordQWdc,wheredcisadeeplyinsightfulpaper(KaneandMele,2005),Kaneandacriticalthickness.ThefirstexperimentalconfirmationoftheMeleshowedin2005thatthestabilitydependsonthenum-existenceoftheQSHstateinHgTe/CdTequantumwellswasberofpairsofedgestates.Anoddnumberofpairsissta-carriedoutbyK¨onigetal.(K¨onigetal.,2007).Thisworkble,whereasanevennumberofpairsisnot.Thisobser-reportstheobservationofanominallyinsulatingstatewhichvationledKaneandMeletoproposeaZ2classificationofconductsonlythrough1Dedgechannels,andisstronglyin- 5fluencedbyaTRsymmetry-breakingmagneticfield.Furthertransportmeasurements(Rothetal.,2009)reporteduniquenonlocalconductionpropertiesduetothehelicaledgestates.TheQSHinsulatorstateisinvariantunderTR,hasachargeexcitationgapinthe2Dbulk,buthastopologicallypro-tected1Dgaplessedgestatesthatlieinsidethebulkinsu-latinggap.Theedgestateshaveadistincthelicalprop-erty:twostateswithoppositespinpolarizationcounter-propagateatagivenedge(KaneandMele,2005;Wuetal.,2006;XuandMoore,2006).Forthisreason,theyarealsocalledhelicaledgestates,i.e.thespiniscorrelatedwiththedirectionofmotion(Wuetal.,2006).TheedgestatescomeinKramersdoublets,andTRsymmetryensuresthecrossingoftheirenergylevelsatspecialpointsintheBrillouinzone.Becauseofthislevelcrossing,thespectrumofaQSHinsu-latorcannotbeadiabaticallydeformedintothatofatopo-logicallytrivialinsulatorwithouthelicaledgestates.There-FIG.2(a)BulkbandstructureofHgTeandCdTe;(b)schematicpic-fore,inthisprecisesense,theQSHinsulatorrepresentsanewtureofquantumwellgeometryandlowestsubbandsfortwodifferenttopologicallydistinctstateofmatter.Inthespecialcasethatthicknesses.FromBernevigetal.,2006.SOCpreservesaU(1)ssubgroupofthefullSU(2)spinrota-tiongroup,thetopologicalpropertiesoftheQSHstatecanbecharacterizedbythespinChernnumber(Shengetal.,2006).2006)(BHZ)todescribethephysicsofthosesubbandsofMoregenerally,thetopologicalpropertiesoftheQSHstateHgTe/CdTequantumwellsthatarerelevantfortheQSHef-aremathematicallycharacterizedbyaZ2topologicalinvari-fect.HgTeandCdTecrystallizeinthezincblendelatticestruc-ant(KaneandMele,2005).Stateswithanevennumberofture.ThisstructurehasthesamegeometryasthediamondKramerspairsofedgestatesatagivenedgearetopologi-lattice,i.e.twointerpenetratingface-centered-cubiclatticescallytrivial,whilethosewithanoddnumberaretopologicallyshiftedalongthebodydiagonal,butwithadifferentatomnontrivial.TheZ2topologicalquantumnumbercanalsobeoneachsublattice.Thepresenceoftwodifferentatomsperdefinedforgenerallyinteractingsystemsandexperimentallylatticesitebreaksinversionsymmetry,andthusreducesthemeasuredintermsofthefractionalchargeandquantizedcur-pointgroupsymmetryfromOh(cubic)toTd(tetrahedral).rentontheedge(Qietal.,2008),andspin-chargeseparationHowever,eventhoughinversionsymmetryisexplicitlybro-inthebulk(QiandZhang,2008;Ranetal.,2008).ken,thisonlyhasasmalleffectonthephysicsoftheQSHInthissection,weshallfocusonthebasictheoryoftheeffect.Tosimplifythediscussion,weshallfirstignorethisQSHstateintheHgTe/CdTesystembecauseofitssimplicitybulkinversionasymmetry(BIA).andexperimentalrelevance,andprovideanexplicitandped-ForbothHgTeandCdTe,theimportantbandsnearagogicaldiscussionofthehelicaledgestatesandtheirtrans-theFermilevelareclosetotheΓpointintheBrillouinportproperties.Thereareseveralothertheoreticalpropos-zone[Fig.2(a)].Theyareas-typeband(Γ6),andap-typealsfortheQSHstate,includingbilayerbismuth(Murakami,bandsplitbySOCintoaJ=3/2band(Γ8)andaJ=1/22006),andthe“broken-gap”type-IIAlSb/InAs/GaSbquan-band(Γ7).CdTehasabandorderingsimilartoGaAswithas-tumwells(Liuetal.,2008).Initialexperimentsinthetype(Γ6)conductionband,andp-typevalencebands(Γ8,Γ7)AlSb/InAs/GaSbsystemalreadyshowencouragingsigna-whichareseparatedfromtheconductionbandbyalargeen-tures(Knezetal.,2010).TheQSHsystemhasalsobeenpro-ergygap(∼1.6eV).BecauseofthelargeSOCpresentinposedforthetransitionmetaloxideNa2IrO3(Shitadeetal.,theheavyelementHg,theusualbandorderingisinverted:2009).TheconceptoffractionalQSHstatewasproposedthenegativeenergygapof−300meVindicatesthattheΓ8atthesametimeastheQSH(BernevigandZhang,2006),band,whichusuallyformsthevalenceband,isabovetheΓ6andhasbeeninvestigatedtheoreticallyinmoredetailsre-band.Thelight-holeΓ8bandbecomestheconductionband,cently(LevinandStern,2009;Youngetal.,2008).theheavy-holebandbecomesthefirstvalenceband,andthes-typeband(Γ6)ispushedbelowtheFermileveltoliebe-tweentheheavy-holebandandthespin-orbitsplit-offbandA.Effectivemodelofthetwo-dimensionaltime-reversal(Γ7)[Fig.2(a)].Duetothedegeneracybetweenheavy-holeinvarianttopologicalinsulatorinHgTe/CdTequantumandlight-holebandsattheΓpoint,HgTeisazero-gapsemi-wellsconductor.WhenHgTe-basedquantumwellstructuresaregrown,theInthissectionwereviewthebasicelectronicstructureofpeculiarpropertiesofthewellmaterialcanbeutilizedtotunebulkHgTeandCdTe,andpresentasimplemodelfirstin-theelectronicstructure.ForwideQWlayers,quantumcon-troducedbyBernevig,HughesandZhang(Bernevigetal.,finementisweakandthebandstructureremains“inverted”. 6onlytermsallowedinthediagonalelementsaretermsthathaveevenpowersofkincludingk-independentterms.Thesubbandsmustcomeindegeneratepairsateachk,sotherecanbenomatrixelementsbetweenthe+stateandthe−stateofthesameband.Finally,iftherearenonzeromatrixelementsbetween|E1+i,|H1−ior|E1−i,|H1+i,thiswouldinduceahigher-orderprocesscouplingthe±statesofthesamebandandsplittingthedegeneracy.Therefore,thesematrixelementsareforbiddenaswell.Thesesimpleargumentsledtothefol-lowingmodel,h(k)0H=∗,(1)0h(−k)FIG.3EnergylevelsoftheQWasafunctionofQWwidth.aFromK¨onigetal.,2008.h(k)=ǫ(k)I2×2+da(k)σ,(2)whereI2×2isthe2×2identitymatrix,andHowever,theconfinementenergyincreaseswhenthewellǫ(k)=C−D(k2+k2),xywidthisreduced.Thus,theenergylevelswillbeshiftedand,da(k)=(Akx,−Aky,M(k)),eventually,theenergybandswillbealignedina“normal”22way,iftheQWthicknessdfallsbelowacriticalthicknessM(k)=M−B(kx+ky),(3)QWdc.Wecanunderstandthisheuristicallyasfollows:forthinwhereA,B,C,D,MarematerialparametersthatdependonQWstheheterostructureshouldbehavesimilarlytoCdTeandtheQWgeometry,andwechoosethezeroofenergytobethehaveanormalbandordering,i.e.thebandswithprimarilyΓ6valencebandedgeofHgTeatk=0[Fig.2].symmetryaretheconductionsubbandsandtheΓ8bandscon-ThebulkenergyspectrumoftheBHZmodelisgivenbytributetothevalencesubbands.Ontheotherhand,asdQWisincreased,weexpectthematerialtobehavemorelikeHgTepE±=ǫ(k)±dada(4)whichhasinvertedbands.AsdQWincreases,weexpecttoqreachacriticalthicknesswheretheΓ8andΓ6subbandscross=ǫ(k)±A2(kx2+ky2)+M2(k).(5)andbecomeinverted,withtheΓ8bandsbecomingconduc-tionsubbandsandtheΓ6bandsbecomingvalencesubbandsForB=0,themodelreducestotwocopiesofthemassive[Fig.2(b)](Bernevigetal.,2006;Noviketal.,2005).TheDiracHamiltonianin(2+1)D.ThemassMcorrespondstoshiftofenergylevelswithdQWisdepictedinFig.3.TheQWtheenergydifferencebetweentheE1andH1levelsattheΓstatesderivedfromtheheavy-holeΓ8bandaredenotedbypoint.ThemassMchangessignatthecriticalthicknessdc,Hn,wherethesubscriptn=1,2,3,...describeswellstateswhereE1andH1becomedegenerate.Atthecriticalpoint,withincreasingnumberofnodesinthezdirection.Similarly,thesystemisdescribedbytwocopiesofthemasslessDiractheQWstatesderivedfromtheelectronΓ6bandaredenotedHamiltonian,oneforeachspin,andatasinglevalleyk=byEn.TheinversionbetweenE1andH1bandsoccursata0.Thissituationissimilartographene(CastroNetoetal.,criticalthicknessdQW=dc∼6.3nm[Fig.3].Inthefollow-2009),whichisalsodescribedbythemasslessDiracHamil-ing,wedevelopasimplemodelanddiscusswhyweexpecttonianin(2+1)D.However,thecrucialdifferenceliesintheQWswithdQW>dctoformTRinvariant2DtopologicalfactthatgraphenehasfourDiraccones,consistingoftwoval-insulatorswithprotectededgestates.leysandtwospins,whereaswehavetwoDiraccones,oneUnderourassumptionofinversionsymmetry,therele-foreachspin,andatasinglevalley.FordQW>dc,theE1vantsubbands,E1andH1,mustbedoublydegeneratesincelevelfallsbelowtheH1levelattheΓpoint,andthemassMTRsymmetryispresent.Weexpressstatesinthebasisbecomesnegative.ApuremassiveDiracmodeldoesnotdif-{|E1+i,|H1+i,|E1−i,|H1−i},where|E1±iand|H1±iferentiatebetweenapositiveornegativemassM.SincewearetwosetsofKramerspartners.Thestates|E1±iand|H1±iaredealingwithanonrelativisticsystem,theBtermisgener-haveoppositeparity,henceaHamiltonianmatrixelementthatallyallowed.Inordertomakethedistinctionclear,wecallMconnectsthemmustbeoddunderparity.Thus,tolowestor-theDiracmass,andBtheNewtonianmass,sinceitdescribesderink,(|E1+i,|H1+i)and(|E1−i,|H1−i)willeachbetheusualnonrelativisticmasstermwithquadraticdispersioncoupledgenericallyviaatermlinearink.The|H1+iheavy-relation.Weshallshowlaterthattherelativesignbetweentheholestateisformedfromthespin-orbitcoupledp-orbitalsDiracmassMandtheNewtonianmassBiscrucialtodeter-|px+ipy,↑i,whilethe|H1−iheavy-holestateisformedfromminewhetherthemodeldescribesatopologicalinsulatorstatethespin-orbitcoupledp-orbitals|−(px−ipy),↓i.Therefore,withprotectededgestatesornot.topreserverotationsymmetryaroundthegrowthaxisz,theHgTehasacrystalstructureofthezincblendetypewhichmatrixelementsmustbyproportionaltok±=kx±iky.Thelacksinversionsymmetry,leadingtoaBIAtermintheHamil- 72d(A)˚A(eV·A)˚B(eV·A˚)D(eV)M(eV)∆z(eV)themodelHamiltonianintotwoparts,553.87−48.0−30.60.0090.0018Hˆ=H˜+H˜,(8)01613.78−55.3−37.8−0.000150.0017M˜(kx)Akx00703.65−68.6−51.2−0.0100.0016H˜=˜ǫ(k)+Akx−M˜(kx)00,0xTABLEIMaterialparametersforHgTe/CdTequantumwellswith00M˜(kx)−Akxdifferentwellthicknessesd.00−Akx−M˜(kx)−Bk2iAk00yytonian,giventoleadingorderby(K¨onigetal.,2008)−iAkBk200H˜=−Dk2+yy,(9)1y00−Bk2iAk000−∆yyz00−iAkBk200∆z0yyHBIA=0∆00.(6)zwithǫ˜(k)=C−Dk2andM˜(k)=M−Bk2.Allk-xxxxx−∆z000dependenttermsareincludedinH˜0.Forsuchasemi-infinitesystem,kxneedstobereplacedbytheoperator−i∂x.OnThistermplaysanimportantroleindeterminingthespintheotherhand,translationsymmetryalongtheydirectionisorientationofthehelicaledgestate.Thetopologicalphasepreserved,sothatkyisagoodquantumnumber.Forky=0,transitioninthepresenceofBIAhasbeeninvestigatedre-cently(K¨onigetal.,2008;Murakamietal.,2007).Inaddi-wehaveH˜1=0andthewaveequationisgivenbytion,inanasymmetricQWstructuralinversionsymmetrycanH˜0(kx→−i∂x)Ψ(x)=EΨ(x).(10)bebrokenbyabuild-inelectricfield,leadingtoaSOCtermofRashbatypeintheeffectiveHamiltonian(Rotheetal.,2010;SinceH˜0isblock-diagonal,theeigenstateshavetheform!!Str¨ometal.,2010).Forsimplicity,wewillfocusonsymmet-ψ00ricQWwitoutSIA.InTableI,wegivetheparametersoftheΨ↑(x)=,Ψ↓(x)=,(11)0ψ0BHZmodelforvariousvaluesofdQW.where0isatwo-componentzerovector,andΨ↑(x)isrelatedtoΨ↓(x)byTR.Fortheedgestates,thewavefunctionψ0(x)islocalizedattheedgeandsatisfiesthewaveequationForthepurposesofstudyingthetopologicalpropertiesof!!thissystem,aswellastheedgestates,itissometimescon-M˜(−i∂x)−iA1∂xǫ˜(−i∂x)+ψ0(x)=Eψ0(12)(x),venienttoworkwithalatticeregularizationofthecontinuum−iA1∂x−M˜(−i∂x)model(1)whichgivestheenergyspectrumovertheentirewhichhasbeensolvedanalyticallyforopenboundaryBrillouinzone,i.e.atight-bindingrepresentation.Sinceallconditionsusingdifferentmethods(K¨onigetal.,2008;theinterestingphysicsatlowenergyoccursneartheΓpoint,Linderetal.,2009;Luetal.,2010;Zhouetal.,2008).Inor-thebehaviorofthedispersionatenergiesmuchlargerthanthedertoshowtheexistenceoftheedgestatesandtofindthere-bulkgapattheΓpointisnotimportant.Thus,wecanchoosegionwheretheedgestatesexist,webrieflyreviewthederiva-aregularizationtosimplifyourcalculations.Thissimplifiedtionoftheexplicitformoftheedgestatesbyneglecting˜ǫforlatticemodelconsistsofreplacing(3)bysimplicity(K¨onigetal.,2008).−2Neglectingǫ˜,thewaveequation(12)hasparticle-holesym-ǫ(k)=C−2Da(2−coskxa−coskya),
metry.Therefore,weexpectthataspecialedgestatewith−1−1da(k)=Aasinkxa,−Aasinkya,M(k),E=0canexist.Withthewavefunctionansatzψ0=φeλx,M(k)=M−2Ba−2(2−coska−coska).(7)theaboveequationcanbesimplifiedtoxy
M+Bλ2τφ=Aλφ,(13)yItisclearthatneartheΓpoint,thelatticeHamiltonianreducestothecontinuumBHZmodelinEq.(1).Forsimplicity,belowthereforethetwo-componentwavefunctionφshouldbeanweworkinunitswherethelatticeconstanta=1.eigenstateofthePaulimatrixτy.Definingatwo-componentspinorφ±byτyφ±=±φ±,Eq.(13)issimplifiedtoaquadraticequationforλ.Ifλisasolutionforφ+,then−λisB.Explicitsolutionofthehelicaledgestatesasolutionforφ−.Consequently,thegeneralsolutionisgivenbyTheexistenceoftopologicallyprotectededgestatesisanψ(x)=(aeλ1x+beλ2x)φ+(ce−λ1x+de−λ2x)φ(14),0+−importantpropertyoftheQSHinsulator.TheedgestatescanbeobtainedbysolvingtheBHZmodel(2)withanopenwhereλ1,2satisfyboundarycondition.ConsiderthemodelHamiltonian(2)de-1pfinedonthehalf-spacex>0inthexyplane.Wecandivideλ1,2=A±A2−4MB.(15)2B 8Thecoefficientsa,b,c,dcanbedeterminedbyimposingthe(a)0.05openboundaryconditionψ(0)=0.Togetherwiththenor-malizabilityofthewavefunctionintheregionx>0,theopenboundaryconditionleadstoanexistenceconditionfortheedgestates:ℜλ1,2<0(c=d=0)orℜλ1,2>00(a=b=0),whereℜstandsfortherealpart.AsseenfromE(k)(eV)Eq.(15),theseconditionscanonlybesatisfiedintheinvertedregimewhenM/B>0.Furthermore,onecanshowthat−0.05whenA/B<0,wehaveℜλ1,2<0,whilewhenA/B>0,−0.02−0.0100.010.02−1k(A)wehaveℜλ1,2>0.Therefore,thewavefunctionfortheedgestatesattheΓpointisgivenby(b)0.05(
aeλ1x−eλ2xφ,A/B<0;+ψ0(x)=
(16)ce−λ1x−e−λ2xφ,A/B>0.−0ThesignofA/BdeterminesthespinpolarizationoftheedgeE(k)(eV)states,whichiskeytodeterminethehelicityoftheDiracHamiltonianforthetopologicaledgestates.Anotherim-portantquantitycharacterizingtheedgestatesistheirdecay−0.05−0.02−0.0100.010.02length,whichisdefinedasl=max|ℜλ|−1.−1c1,2k(A)TheeffectiveedgemodelcanbeobtainedbyprojectingthebulkHamiltonianontotheedgestatesΨ↑andΨ↓definedinEq.(11).Thisprocedureleadstoa2×2effectiveHamil-FIG.4EnergyspectrumoftheeffectiveHamiltonian(2)inacylin-toniandefinedbyHαβ(k)=hΨ|H˜+H˜|Ψi.Todergeometry.InathinQW,(a)thereisagapbetweenconductionedgeyα01βbandandvalenceband.InathickQW,(b)therearegaplessedgeleadingorderinky,wearriveattheeffectiveHamiltonianforstatesontheleftandrightedge(redandbluelines,respectively).thehelicaledgestates:Thedashedlinestandsforatypicalvalueofthechemicalpotentialwithinthebulkgap.AdaptedfromQiandZhang,2010.zHedge=Akyσ.(17)ForHgTeQWs,wehaveA≃3.6eV·A(K¨onig˚etal.,2008),IntheQHeffect,thechiraledgestatescannotbebackscat-andtheDiracvelocityoftheedgestatesisgivenbyv=teredforsamplewidthslargerthanthedecaylengthoftheA/~≃5.5×105m/s.edgestates.IntheQSHeffect,onemaynaturallyaskwhetherTheanalyticalcalculationabovecanbeconfirmedbyexactbackscatteringofthehelicaledgestatesispossible.ItturnsnumericaldiagonalizationoftheHamiltonian(2)onastripoutthatTRsymmetrypreventsthehelicaledgestatesfromoffinitewidth,whichcanalsoincludethecontributionofbackscattering.Theabsenceofbackscatteringreliesonthetheǫ(k)term[Fig.4].Thefinitedecaylengthoftheheli-destructiveinterferencebetweenallpossiblebackscatteringcaledgestatesintothebulkdeterminestheamplitudeforin-pathstakenbytheedgeelectrons.teredgetunneling(Houetal.,2009;Str¨omandJohannesson,Beforegivingasemiclassicalargumentwhythisisso,2009;Tanakaetal.,2009;TeoandKane,2009;Zhouetal.,wefirstconsiderananalogyfromdailyexperience.Most2008;ZyuzinandFiete,2010).eyeglassesandcameralenseshaveanantireflectivecoating[Fig.5(a)],wherelightreflectedfromthetopandbottomsur-facesinterferedestructively,leadingtononetreflectionandC.Physicalpropertiesofthehelicaledgestatesthusperfecttransmission.However,thiseffectisnotrobust,asitdependsonaprecisematchingbetweenthewavelength1.Topologicalprotectionofthehelicaledgestatesoflightandthethicknessofthecoating.Nowweturntothehelicaledgestates.IfanonmagneticimpurityispresentnearFromtheexplicitanalyticalsolutionoftheBHZmodel,theedge,itcaninprinciplecausebackscatteringoftheheli-thereisapairofhelicaledgestatesexponentiallylocalizedcaledgestatesduetoSOC.However,justasforthereflec-attheedge,anddescribedbytheeffectivehelicaledgethe-tionofphotonsbyasurface,anelectroncanbereflectedbyory(17).Inthiscontext,theconceptof“helical”edgeanonmagneticimpurity,anddifferentreflectionpathsinter-state(Wuetal.,2006)referstothefactthatstateswithop-ferequantum-mechanically.Aforward-movingelectronwithpositespincounter-propagateatagivenedge,asweseefromspinupontheQSHedgecanmakeeitheraclockwiseoratheedgestatedispersionrelationshowninFig.4(b),orthecounterclockwiseturnaroundtheimpurity[Fig.5(b)].SincerealspacepictureshowninFig.1(b).Thisisinsharpcontrastonlyspindownelectronscanpropagatebackwards,theelec-tothe“chiral”edgestatesintheQHstate,wheretheedgetronspinhastorotateadiabatically,eitherbyanangleofπstatespropagateinonedirectiononly,asshowninFig.1(a).or−π,i.e.intotheoppositedirection.Consequently,thetwo 9Kramers'theorem.IfweaddTRinvariantperturbationstotheabHamiltonian,wecanmovethedegeneratepointupanddowninenergy,butcannotremovethedegeneracy.Inthisprecisesense,thehelicaledgestatesaretopologicallyprotectedbyTRsymmetry.IfTRsymmetryisnotpresent,asimple“mass”termcanbeaddedtotheHamiltoniansothatthespectrumbecomesgapped:Zdk†Hmass=mψk+ψk−+h.c.,2π†whereh.c.denotesHermitianconjugation,andψk±,ψk±areFIG.5(a)Onalenswithantireflectivecoating,lightreflectedbycreation/annihilationoperatorsforanedgeelectronofmo-top(blueline)andbottom(redline)surfacesinterferesdestructively,mentumk,with±denotingtheelectronspin.Theactionofleadingtosuppressedreflection.(b)TwopossiblepathstakenbyanTRsymmetryontheelectronoperatorsisgivenbyelectrononaQSHedgewhenscatteredbyanonmagneticimpurity.◦Theelectronspinrotatesby180clockwisealongthebluecurve,and−1−1Tψk+T=ψ−k,−,Tψk−T=−ψ−k,+,(18)counterclockwisealongtheredcurve.Ageometricalphasefactorassociatedwiththisrotationofthespinleadstodestructiveinterfer-whichimpliesencebetweenthetwopaths.Inotherwords,electronbackscatteringontheQSHedgeissuppressedinawaysimilartohowthereflec-THT−1=−H.massmasstionofphotonsissuppressedbyanantireflectivecoating.AdaptedfromQiandZhang,2010.Consequently,HmassisaTRsymmetrybreakingperturba-tion.Moregenerally,ifwedefinethe“chirality”operatorZpathsdifferbyafullπ−(−π)=2πrotationoftheelectrondk††spin.However,thewavefunctionofaspin-1/2particlepicksC=N+−N−=ψk+ψk+−ψk−ψk−,2πupanegativesignunderafull2πrotation.Therefore,twoanyoperatorthatchangesCby2(2n−1),n∈ZisbackscatteringpathsrelatedbyTRalwaysinterferedestruc-oddunderTR.Inotherwords,TRsymmetryonlyallowstively,leadingtoperfecttransmission.Iftheimpuritycarries2n-particlebackscattering,describedbyoperatorssuchasamagneticmoment,TRsymmetryisexplicitlybroken,and††thetworeflectedwavesnolongerinterferedestructively.Inψk+ψk′+ψp−ψp′−(forn=1).Therefore,themostrelevant†thisway,therobustnessoftheQSHedgestateisprotectedbyperturbationψk+ψk′−isforbiddenbyTRsymmetry,whichisTRsymmetry.essentialforthetopologicalstabilityoftheedgestates.ThisThephysicalpicturedescribedaboveappliesonlytotheedgestateeffectivetheoryisnonchiral,andisqualitativelycaseofasinglepairofQSHedgestates(KaneandMele,differentfromtheusualspinlessorspinfulLuttingerliquid2005;Wuetal.,2006;XuandMoore,2006).Iftherearetheories.Itcanbeconsideredasanewclassof1Dcriticaltwoforward-moversandtwobackward-moversonagiventheories,dubbeda“helicalliquid”(Wuetal.,2006).Specifi-edge,anelectroncanbescatteredfromaforward-movingtocally,inthenoninteractingcasenoTRinvariantperturbationabackward-movingchannelwithoutreversingitsspin.Thisisavailabletoinducebackscattering,sothattheedgestateisspoilstheperfectdestructiveinterferencedescribedabove,robust.andleadstodissipation.Consequently,fortheQSHstatetoConsidernowthecaseoftwoflavorsofhelicaledgestatesberobust,edgestatesmustconsistofanoddnumberoffor-ontheboundary,i.e.a1Dsystemconsistingoftwoleft-ward(backward)movers.Thiseven-oddeffectisthekeyrea-moversandtworight-moverswithHamiltoniansonwhytheQSHinsulatorischaracterizedbyaZ2topolog-ZdkX††icalquantumnumber(KaneandMele,2005;Wuetal.,2006;H=ψks+vkψks+−ψks−vkψks−.2πXuandMoore,2006).s=1,2ThegeneralpropertiesofTRsymmetryareimportantforRAmasstermsuchasm˜dkψ†ψ−ψ†ψ+h.c.understandingthepropertiesoftheedgetheory.Theanti-2πk1+k2−k1−k2+unitaryTRoperatorTtakesdifferentformsdependingon(withm˜real)canopenagapinthesystemwhilepreservingwhetherthedegreesoffreedomhaveintegerorhalf-odd-time-reversalsymmetry.Inotherwords,twocopiesoftheintegerspin.Forhalf-odd-integerspin,wehaveT2=−1helicalliquidformaatopologicallytrivialtheory.Morewhichimplies,byKramers'theorem,thatanysingle-particlegenerally,anedgesystemwithTRsymmetryisanontrivialeigenstateoftheHamiltonianmusthaveadegeneratepart-helicalliquidwhenthereisanoddnumberofleft-(right-)ner.FromFig.4(b),weseethatthetwodispersionbranchesmovers,andtrivialwhenthereisanevennumberofthem.atonegivenedgecrosseachotherattheTRinvariantk=0ThusthetopologyofQSHsystemsarecharacterizedbyaZ2point.Atthispoint,thesetwodegeneratestatesexactlysatisfytopologicalquantumnumber. 102.InteractionsandquencheddisorderK<3/8(GiamarchiandSchulz,1988;Wuetal.,2006;XuandMoore,2006).AtT=0,Nx,y(x)exhibitsglassyWenowreviewtheeffectofinteractionsandquenchedbehavior,i.e.disorderedinthespatialdirectionbutstaticindisorderontheQSHedgeliquid(Wuetal.,2006;thetimedirection.SpintransportisthusblockedandTRXuandMoore,2006).OnlytwoTRinvariantnonchiralsymmetryisagainspontaneouslybrokenatT=0.AtlowinteractionscanbeaddedtoEq.17,theforwardandUmklappbutfiniteT,thesystemremainsgappedwithTRsymmetryscatteringsrestored.ZIntheabove,wehaveseenthatthehelicalliquidcanin††Hf=gdxψ+ψ+ψ−ψ−(19)principlebedestroyed.However,forareasonablyweakinter-Zactingsystem,i.e.K≈1,theone-componenthelicalliquidH=gdxe−i4kFxψ†(x)ψ†(x+a)remainsgapless.InanIsingorderedphase,thelow-energyex-uu++citationsontheedgeareIsingdomainwallswhichcarryfrac-×ψ−(x+a)ψ−(x)+h.c.,(20)tionale/2charge(Qietal.,2008).Thepropertiesofmulti-††componenthelicalliquidsinthepresenceofdisorderhasalsowherethetwo-particleoperatorsψψ,ψψarepoint-splitbeenstudied(XuandMoore,2006).withthelatticeconstantawhichplaystheroleofashort-AmagneticimpurityontheedgeofaQSHinsulatorisex-distancecutoff.Thechiralinteractiontermsonlyrenormalizepectedtoactasalocalmasstermfortheedgetheory,andtheFermivelocityv,andarethusignored.Itiswellknownthusisexpectedtoleadtoasuppressionoftheedgeconduc-thattheforwardscatteringtermgivesanontrivialLuttingpertance.Whilethisiscertainlytrueforastaticmagneticim-parameterK=(v−g)/(v+g),butkeepsthesystemgap-purity,aquantummagneticimpurity,i.e.aKondoimpurity,less.OnlytheUmklapptermhasthepotentialtoopenupagapleadstosubtlerbehavior(Maciejkoetal.,2009;Wuetal.,atthecommensuratefillingkF=π/2.Thebosonizedform2006).Inthepresenceofaquantummagneticimpurity,dueoftheHamiltonianreadsZ√tothecombinedeffectsofinteractionsandSOConemustnov¯122gucos16πφalsogenerallyconsiderlocaltwo-particlebackscatteringpro-H=dx(∂xφ)+K(∂xθ)+,2K2(πa)2cesses(MeidanandOreg,2005)similartoEq.(20),butoc-(21)curringonlyatthepositionoftheimpurity.Athightem-pperatures,bothweakKondoandweaktwo-particlebackscat-wherev¯=v2−g2istherenormalizedvelocity,andweteringareexpectedtogiverisetoalogarithmictempera-definenonchiralbosonsφ=φR+φLandθ=φR−φL,turedependenceasintheusualKondoeffect(Maciejkoetal.,respectively,whereφRandφLarechiralbosonsdescrib-2009),andtheireffectisnoteasilydistinguishable.How-ingthespinup(down)right-moverandthespindown(up)ever,atlowtemperaturesthephysicsdependsdrasticallyonleft-mover,respectively.φcontainsbothspinandchargethestrengthofCoulombinteractionsontheedge,parameter-degreesoffreedom,andisequivalenttothecombinationizedbytheLuttingerparameterK.ForweakCoulombin-φc−θsinthespinfulLuttingerliquid,withφcandθstheteractionsK>1/4,theedgeconductanceisrestoredtothechargeandspinbosons,respectively(Giamarchi,2003).Itunitaritylimit2e2/hwithunusualpowerlawscharacteristic√isalsoacompactvariablewithperiodπ.Arenormaliza-ofa“localhelicalliquid”(Maciejkoetal.,2009;Wuetal.,tiongroupanalysisshowsthattheUmklapptermisrelevant2006).ForstrongCoulombinteractionsK<1/4,thecon-forK<1/2withapinnedvalueofφ.Consequently,agapductancevanishesatT=0,butisrestoredatlowTby1∆∼a−1(g)2−4Kopensandspintransportisblocked.Theuafractionalizedtunnelingcurrentofchargee/2quasiparti-massorderparameters√Nx,ythebosonizedformofwhichis√cles(Maciejkoetal.,2009).Thetunnelingofachargee/2iηRηLiηRηLNx=2πasin4πφ,Ny=2πacos4πφ√,areoddun-quasiparticleisdescribedbyaninstantonprocesswhichisthederTR.Forgu<0,φispinnedateither0orπ/2,andthetimecounterparttothestatice/2chargeonaspatialmagneticNyorderisIsing-like.AtT=0,thesystemisinaIsingor-domainwallalongtheedge(Qietal.,2008).Inadditiontothederedphase,andTRsymmetryisspontaneouslybroken.Onsingle-channelKondoeffectjustdescribed,thepossibilityoftheotherhand,when00whereNxis2010).theorderparameter.Thereisalsothepossibilityoftwo-particlebackscatteringduetoquencheddisorder,describedbytheterm3.HelicaledgestatesandtheholographicprincipleZgu(x)√Hdis=dxcos16π(φ(x,τ)+α(x)),(22)Thereisanalternativewaytounderstandthequalitative2(πa)2differencebetweenanevenandoddnumberofedgestateswherethescatteringstrengthgu(x)andphaseα(x)intermsofa“fermiondoubling”theorem(Wuetal.,2006).areGaussianrandomvariables.ThestandardreplicaThistheoremstatesthatthereisalwaysanevennumberofanalysisshowsthatdisorderbecomesrelevantatKramerspairsattheFermienergyforaTRinvariant,but 11numberofKramerspairs,thecouplingbetweenthemcanonly(a)(b)annihilateanevennumberofKramerspairsifTRispreserved.EEAsaresult,atleastonepairofgaplessedgestatescansurvive.Thisfermiondoublingtheoremcanbegeneralizedto3DeFeFinastraightforwardway.Inthe2DQSHstate,thesimplesthelicaledgestateconsistsofasinglemasslessDiracfermionin(1+1)D.Thesimplest3Dtopologicalinsulatorcontains-p0p-p0pasurfacestateconsistingofasinglemasslessDiracfermionin(2+1)D.AsinglemasslessDiracfermionwouldalsoFIG.6(a)Energydispersionofa1DTRinvariantsystem.TheviolatethefermiondoublingtheoremandcannotexistinaKramersdegeneracyisrequiredatk=0andk=π,sothatthepurely2DsystemwithTRsymmetry.However,itcanex-energyspectrumalwayscrosses4ntimestheFermilevelǫF.(b)istholographically,astheboundaryofa3Dtopologicalin-EnergydispersionofthehelicaledgestatesononeboundaryoftheQSHsystem(solidlines).Atk=0theedgestatesareKramerssulator.Moregenerically,thereisaone-to-onecorrespon-partners,whileatk=πtheymergeintothebulkandpairwithdencebetweentopologicalinsulatorsandrobustgaplessthe-theedgestatesoftheotherboundary(dashlines).Inboth(a)andoriesinonelowerdimension(Freedmanetal.,2010;Kitaev,(b),redandbluelinesrepresentthetwopartnersofaKramerspair.2009;TeoandKane,2010)FromK¨onigetal.,2008.4.Transporttheoryofthehelicaledgestatesotherwisearbitrary1Dbandstructure.Asinglepairofhe-licalstatescanoccuronly“holograhically”,i.e.whentheInconventionaldiffusiveelectronics,bulktransportsat-1Dsystemistheboundaryofa2Dsystem.ThisfermionisfiesOhm'slaw.ResistanceisproportionaltothelengthdoublingtheoremisaTRinvariantgeneralizationoftheandinverselyproportionaltothecross-sectionalarea,im-Nielsen-Ninomiyano-gotheoremforchiralfermionsonalat-plyingtheexistenceofalocalresistivityorconductivitytice(NielsenandNinomiya,1981).Forspinlessfermions,tensor.However,insystemssuchastheQHandQSHthereisalwaysanequalnumberofleft-moversandright-states,theexistenceofedgestatesnecessarilyleadstonon-moversattheFermilevel,whichleadstothefermiondou-localtransportwhichinvalidatestheconceptoflocalresis-blingprobleminoddspatialdimensions.Ageometricalwaytivity.Suchnonlocaltransporthasbeenexperimentallyob-tounderstandthisresultisthatforperiodicfunctions(i.e.en-servedintheQHregimeinthepresenceofalargemag-ergyspectraofalatticemodel),“whatgoesupmusteventuallyneticfield(BeenakkerandvanHouten,1991),andthenon-comedown”.Similarly,foraTRsymmetricsystemwithhalf-localtransportiswelldescribedbyaquantumtransportodd-integerspins,Kramers'theoremrequiresthateacheigen-theorybasedontheLandauer-B¨uttikerformalism(B¨uttiker,stateoftheHamiltonianisaccompaniedbyitsTRconjugate1988).AsimilartransporttheoryhasbeendevelopedfororKramerspartner,sothatthenumberoflow-energychannelsthehelicaledgestatesoftheQSHstates,andthenonlo-isdoubled.AKramerspairofstatesatk=0mustrecombinecaltransportexperimentsareinexcellentagreementwithintopairswhenkgoesfrom0toπand2π,whichrequiresthethetheory(Rothetal.,2009).ThesemeasurementsarenowbandstocrosstheFermilevel4ntimes[Fig.6(a)].However,widelyacknowledgedasconstitutingdefinitiveexperimen-thereisanexceptiontothistheorem,whichisanalogoustothetalevidencefortheexistenceofedgestatesintheQSHreasonwhyachiralliquidcanexistintheQHeffect.Ahelicalregime(B¨uttiker,2009).liquidwithanoddnumberoffermionbranchescanoccurifWithinthegeneralLandauer-B¨uttikerformalism(B¨uttiker,itisholographic,i.e.ifitappearsattheboundary(edge)ofa1986),thecurrent-voltagerelationshipisexpressedas2Dsystem.Inthiscase,theedgestatesareKramerspartnerse2Xatk=0,butmergeintothebulkatsomefinitekc,suchthatIi=h(TjiVi−TijVj),(23)theydonothavetobecombinedatk=π.Moreaccurately,jtheedgestatesonbothleftandrightboundariesbecomesbulkwhereIiisthecurrentflowingoutoftheithelectrodeintothestatesfork>kcandformaKramerspair[Fig.6(b)].Thissampleregion,Viisthevoltageontheithelectrode,andTjiisisexactlythebehaviordiscussedinSec.II.Binthecontextofthetransmissionprobabilityfromtheithtothejthelectrode.Ptheanalyticalsolutiontheedgestatewavefunctions.ThetotalcurrentisconservedinthesensethatiIi=0.Thefermiondoublingtheoremalsoprovidesaphysicalun-Avoltageleadjisdefinedbytheconditionthatitdrawsnoderstandingofthetopologicalstabilityofthehelicalliquid.netcurrent,i.e.Ij=0.Thephysicalcurrentsremainun-Anylocalperturbationontheboundaryofa2DQSHsys-changedifthevoltagesonallelectrodesareshiftedbyacon-PPtemisequivalenttotheactionofcouplinga“dirtysurfacestantamountµ,implyingthatiTij=iTji.InaTRlayer”totheunperturbedhelicaledgestates.Whateverper-invariantsystem,thetransmissioncoefficientssatisfythecon-turbationisconsidered,the“dirtysurfacelayer”isalways1D,ditionTij=Tji.suchthatthereisalwaysanevennumberofKramerspairsofForageneral2Dsample,thenumberoftransmissionchan-low-energychannels.Sincethehelicalliquidhasonlyanoddnelsscaleswiththewidthofthesample,sothatthetransmis- 12sionmatrixTijiscomplicatedandnonuniversal.However,thevoltageleads,theyinteractwithareservoircontainingaatremendoussimplificationarisesifthequantumtransportislargenumberoflow-energydegreesoffreedom,andTRsym-entirelydominatedbytheedgestates.IntheQHregime,chi-metryiseffectivelybrokenbythemacroscopicirreversibility.raledgestatesareresponsibleforthetransport.ForastandardAsaresult,thetwocounter-propagatingchannelsequilibrateHallbarwithNcurrentandvoltageleadsattached,thetrans-atthesamechemicalpotential,determinedbythevoltageofmissionmatrixelementsfortheν=1QHstatearegiventhelead.Dissipationoccurswiththeequilibrationprocess.byT(QH)i+1,i=1,fori=1,...,N,andallotherma-Thetransportequation(23)breaksthemacroscopicTRsym-trixelementsvanishidentically.Hereweperiodicallyiden-metry,eventhoughthemicroscopicTRsymmetryisensuredtifythei=N+1electrodewithi=1.ChiraledgestatesbytherelationshipTij=Tji.Incontrasttothecaseoftheareprotectedfrombackscattering,therefore,theithelectrodeQHstate,theabsenceofdissipationintheQSHhelicaledgetransmitsperfectlytotheneighboring(i+1)thelectrodeonstatesisprotectedbyKramers'theorem,whichreliesontheonesideonly.Intheexampleofcurrentleadsontheelec-quantumphasecoherenceofwavefunctions.Thus,dissipa-trodes1and4,andvoltageleadsontheelectrodes2,3,5andtioncanoccuroncephasecoherenceisdestroyedinthemetal-6,(seetheinsetofFig.12forthelabeling),onefindsthatlicleads.Onthecontrary,therobustnessofQHchiraledgeI=−I≡I,V−V=0andV−V=hI,givingstatesdoesnotrequirephasecoherence.Amorerigorousand14142314e214afour-terminalresistanceofR14,23=0andatwo-terminalmicroscopicanalysisofthedifferentroleplayedbyametallicresistanceofR=h.leadinQHandQSHstateshasbeenperformed(Rothetal.,14,14e2Thehelicaledgestatescanbeviewedastwocopiesofchi-2009),theresultofwhichagreeswiththesimpletransportraledgestatesrelatedbyTRsymmetry.Therefore,thetrans-equations(23)and(24).Thesetwoequationscorrectlyde-missionmatrixisgivenbyT(QSH)=T(QH)+T†(QH),scribethedissipationlessquantumtransportinsidetheQSHimplyingthattheonlynonvanishingmatrixelementsaregiveninsulator,andthedissipationinsidetheelectrodes.AsshownbyinSec.II.F.4,theseequationscanbeputtomorestringentexperimentaltests.T(QSH)i+1,i=T(QSH)i,i+1=1.(24)TheuniquehelicaledgestatesoftheQSHstatecanbeusedtoconstructdeviceswithinterestingtransportproper-Consideringagaintheexampleofcurrentleadsontheelec-ties(Akhmerovetal.,2009;Kharitonov,2010;Zhangetal.,trodes1and4,andvoltageleadsontheelectrodes2,3,52009).Besidestheedgestatetransport,theQSHstatealsoand6,onefindsthatI=−I≡I,V−V=hI1414232e214leadstointerestingbulktransportproperties(Noviketal.,andV−V=3hI,givingafour-terminalresistanceof14e2142010).R=handatwo-terminalresistanceofR=3h.14,232e214,142e2Fourterminalresistancewithdifferentconfigurationsofvolt-ageandcurrentprobescanbepredictedinthesameway,D.Topologicalexcitationswhichareallrationalfractionsofh/e2.Theexperimentaldata[Fig.16]neatlyconfirmsallthesehighlynontrivialtheoreticalIntheprevioussections,wediscussedthetransportprop-predictions(Rothetal.,2009).FortwomicroHallbarstruc-ertiesofthehelicaledgestatesintheQSHstate.UnliketheturesthatdifferonlyinthedimensionsoftheareabetweencaseoftheQHstate,thesetransportpropertiesarenotex-thevoltagecontacts3and4,theexpectedresistancevaluespectedtobepreciselyquantized,sincetheyarenotdirectlyR=handR=3hareindeedobservedforgaterelatedtotheZtopologicalinvariantwhichcharacterizesthe14,232e214,142e22voltagesforwhichthesamplesareintheQSHregime.topologicalstate.Inthissection,weshowthatitispossi-Asmentionedearlier,onemightsenseaparadoxbetweenbletomeasuretheZ2topologicalquantumnumberdirectlythedissipationlessnatureoftheQSHedgestatesandthefiniteinexperiments.Weshalldiscusstwoexamples.Thefirstisfour-terminallongitudinalresistanceR14,23,whichvanishesthefractionalchargeandquantizedcurrentexperimentsattheintheQHstate.WecangenerallyassumethatthemicroscopicedgeofaQSHsystem(Qietal.,2008).Second,wediscussHamiltoniangoverningthevoltageleadsisinvariantunderthespin-chargeseparationeffectoccurringinthebulkoftheTRsymmetry.Therefore,onewouldnaturallyaskhowsuchsample(QiandZhang,2008;Ranetal.,2008).leadscouldcausethedissipationofthehelicaledgestates,whichareprotectedformbackscatteringbyTRsymmetry?Innature,TRsymmetrycanbebrokenintwoways,eitherat1.FractionalchargeontheedgethelevelofthemicroscopicHamiltonian,oratthelevelofthemacroscopicirreversibilityinsystemswhosemicroscopicThefirsttheoreticalproposalwediscussisthatofalocal-HamiltonianrespectsTRsymmetry.WhenthehelicaledgeizedfractionalchargeattheedgeofaQSHsamplewhenstatespropagatewithoutdissipationinsidetheQSHinsulatoramagneticdomainwallispresent.Theconceptoffrac-betweentheelectrodes,neitherformsofTRsymmetrybreak-tionalchargeinacondensedmattersysteminducedatamassingarepresent.Asaresult,thetwocounter-propagatingchan-domainwallgoesbacktotheSu-Schrieffer-Heeger(SSH)nelscanbemaintainedattwodifferentquasi-chemicalpoten-model(Suetal.,1979).Forspinlessfermions,amassdo-tials,leadingtoanetcurrentflow.However,oncetheyentermainwallinducesalocalizedstatewithone-halfoftheelec- 13troncharge.However,forarealmaterialsuchaspoly-Aacetylene,twospinorientationsarepresentforeachelec-mtron,andbecauseofthisdoubling,adomainwallinpoly-xacetyleneonlycarriesintegercharge.Thebeautifulpro-posalofSSH,anditscounterpartinfieldtheory,theJackiw-Rebbimodel(JackiwandRebbi,1976),haveneverbeenex-perimentallyrealized.Asmentionedearlier,conventional1DBelectronicsystemshavefourbasicdegreesoffreedom,i.e.qforward-andbackward-moverswithtwospins.However,aCurrentFlowxhelicalliquidatagivenedgeoftheQSHinsulatorhasonlytwo:aspinup(down)forward-moverandaspindown(up)backward-mover.Therefore,thehelicalliquidhashalfthede-greesoffreedomofaconventional1Dsystem,andthusavoidsFIG.7(a)Schematicpictureofthehalf-chargeonadomainwall.Thebluearrowsshowamagneticdomainwallconfigurationandthethedoublingproblem.Becauseofthisfundamentaltopologi-purplelineshowsthemasskink.Theredcurveshowsthechargeden-calpropertyofthehelicalliquid,adomainwallcarrieschargesitydistribution.(b)Schematicpictureofthepumpinginducedbye/2.Inaddition,ifthemagnetizationisrotatedperiodically,atherotationofmagneticfield.Thebluecirclewitharrowshowsthequantizedchargecurrentwillflow.Thisprovidesadirectre-rotationofthemagneticfieldvector.AdaptedfromQietal.,2008.alizationoftheThoulesstopologicalpump(Thouless,1983).WebeginwiththeedgeHamiltoniangiveninEq.(17).Thesehelicalfermionstatesonlyhavetwodegreesoffree-wallofθ[Fig.7(a)](JackiwandRebbi,1976).Similarly,thedom;thespinpolarizationiscorrelatedwiththedirec-chargepumpedbyapurelytime-dependentθ(t)fieldinatimetionofmotion.Amassterm,beingproportionaltotheinterval[t,t]is∆Q|t2=[θ(t)−θ(t)]/2π.Whenθ12pumpt121Paulimatricesσ1,2,3,canonlybeintroducedintheHamil-isrotatedfrom0to2πadiabatically,aquantizedchargeeistonianbycouplingtoaTRsymmetrybreakingexternalpumpedthroughthe1Dsystem[Fig.7(b)].fieldsuchasamagneticfield,alignedmagneticimpuri-Fromthelinearrelationma=taiBi,theangleθcanbeties(Gaoetal.,2009),orinteraction-drivenferromagneticor-determinedforagivenmagneticfieldB.Independentfromderontheedge(Kharitonov,2010).Toleadingorderinper-thedetailsoftai,oppositemagneticfieldsBand−Balwaysturbationtheory,amagneticfieldgeneratesthemasstermscorrespondtooppositemass,sothatθ(B)=θ(−B)+π.ZXThusthechargelocalizedonananti-phasemagneticdomain†aHM=dxΨma(x,t)σΨwallofmagnetizationfieldisalwayse/2mode,whichisaZa=1,2,3directmanifestationoftheZ2topologicalquantumnumberofXtheQSHstate.Suchahalfchargeisdetectableinaspecially=dxΨ†tB(x,t)σaΨ,(25)aiidesignedsingle-electrontransistordevice(Qietal.,2008).a,iTwhereΨ=(ψ+,ψ−)andthemodel-dependentcoefficientmatrixtaiisdeterminedbythecouplingoftheedgestatesto2.Spin-chargeseparationinthebulkthemagneticfield.AccordingtotheworkofGoldstoneandWilczek(GoldstoneandWilczek,1981),atzerotemperatureInadditiontothefractionalchargeontheedge,therehavetheground-statechargedensityj0≡ρandcurrentj1≡jinbeentheoreticalproposalsforabulkspin-chargeseparationabackgroundfieldma(x,t)isgivenbyeffect(QiandZhang,2008;Ranetal.,2008).TheseideasaresimilartotheZ2spinpumpproposedin(FuandKane,2006).11µναβWefirstpresentanargumentwhichisphysicallyintuitive,butjµ=√αǫǫmα∂νmβ,α,β=1,2,2πmαmonlyvalidwhenthereisatleastaU(1)sspinrotationsymme-withµ,ν=0,1correspondingtothetimeandspacetry,e.g.whenSzisconserved.Inthiscase,theQSHeffectiscomponents,respectively,andm3doesnotenterthelong-simplydefinedastwocopiesoftheQHeffect,withoppositeHallconductancesof±e2/hforoppositespinorientations.wavelengthcharge-responseequation.Ifweparameterizethemasstermsintermsofanangularvariableθ,i.e.m1=Withoutlossofgenerality,wefirstconsideradiskgeometrymcosθ,m2=msinθ,theresponseequationissimplifiedwithanelectromagneticgaugefluxofφ↑=φ↓=hc/2e,ortosimplyπinunitsof~=c=e=1,throughaholeatthecenter[Fig.8].Thegaugefluxactsonbothspinorientations,11ρ=∂xθ(x,t),j=−∂tθ(x,t).(26)andtheπfluxpreservesTRsymmetry.Weconsideradia-2π2πbaticprocessesφ↑(t)andφ↓(t),whereφ↑(t)=φ↓(t)=0Sucharesponseistopologicalinthesensethatthenetchargeatt=0,andφ↑(t)=φ↓(t)=±πatt=1.SincethefluxQinaregion[x1,x2]attimetdependsonlyontheboundaryofπisequivalenttothefluxof−π,therearefourdifferentvaluesofθ(x,t)i.e.Q=[θ(x2,t)−θ(x1,t)]/2π.Inpar-adiabaticprocessesallreachingthesamefinalfluxconfigura-ticular,ahalf-charge±e/2iscarriedbyananti-phasedomaintion.Inprocess(a),φ↑(t)=−φ↓(t)andφ↑(t=1)=π.In 14process(b),φ↑(t)=−φ↓(t)andφ↑(t=1)=−π.Inpro-andthechargequantumnumbersaresharplydefinedquan-cess(c),φ↑(t)=φ↓(t)andφ↑(t=1)=π.Inprocess(d),tumnumbers(KivelsonandSchrieffer,1982).Theinsulatingφ↑(t)=φ↓(t)andφ↑(t=1)=−π.Thesefourprocessesstatehasabulkgap∆,andanassociatedcoherencelengthareillustratedinFig.8.Processes(a)and(b)preserveTRξ∼A/∆whereAistheDiracparameterinEq.1.Aslongsymmetryatallintermediatestages,whileprocesses(c)andastheradiusoftheGaussianlooprGfarexceedsthecoher-(d)onlypreserveTRsymmetryatthefinalstage.encelength,i.e.,rG≫ξ,thespinandthechargequantumWeconsideraGaussianloopsurroundingtheflux.Asthenumbersaresharplydefinedwithexponentialaccuracy.fluxφ↑(t)isturnedonadiabatically,Faraday'slawofinduc-WhenthespinrotationsymmetryisbrokenbutTRsymme-tionstatesthatatangentialelectricfieldE↑isinducedalongtryisstillpresent,theconceptofspin-chargeseparationisstilltheGaussianloop.ThequantizedHallconductanceimpliesawelldefined(QiandZhang,2008).Aspinonstatecanbede-2radialcurrentj=eˆz×E,resultinginanetchargeflowfinedasaKramersdoubletwithoutanycharge,andaholonor↑h↑∆Q↑throughtheGaussianloop:achargeonisaKramerssingletcarryingcharge±e.Bycom-Z1ZZ1Zbiningthespinandchargefluxthreading(EssinandMoore,e22007),itcanbeshowngenerallythatthesespin-charge∆Q↑=−dtdn·j↑=−dtdl·E↑0h0separatedquantumnumbersarelocalizednearaφ=πe2Z1∂φe2hceflux(QiandZhang,2008;Ranetal.,2008).=−dt=−=−.(27)hc0∂thc2e2AnidenticalargumentappliedtothespindowncomponentE.QuantumanomalousHallinsulatorshowsthat∆Q↓=−e/2.Therefore,thisadiabaticprocesscreatestheholonstatewith∆Q=∆Q↑+∆Q↓=−eandAlthoughTRinvarianceisessentialintheQSHinsulator,∆Sz=∆Q↑−∆Q↓=0.thereisaTRsymmetrybreakingstateofmatterwhichiscloselyrelatedtotheQSHinsulator:thequantumanomalousHall(QAH)insulator.TheQAHinsulatorisabandinsula-torwithquantizedHallconductancebutwithoutorbitalmag-neticfield.Nearlytwodecadesago,Haldane(Haldane,1988)proposedamodelonahoneycomblatticewheretheQHisrealizedwithoutanyexternalmagneticfield,orthebreakingoftranslationalsymmetry.However,themicroscopicmecha-nismofcirculatingcurrentloopswithinoneunitcellhasnotbeenrealizedinanymaterials.Qi,WuandZhang(Qietal.,2006)proposedasimplemodelbasedontheconceptoftheQAHinsulatorwithferromagneticmomentsinteractingwithbandelectronsviatheSOC.Thissimplemodelcanbereal-izedinrealmaterials.Tworecentproposals(Liuetal.,2008;Yuetal.,2010)makeuseofthepropertiesofTRinvarianttopologicalinsulatorstorealizetheQAHstatebymagneticdoping.Thisisnotaccidental,butshowsthedeeprelation-shipbetweenthesetwostatesofmatter.ThuswegiveabriefFIG.8Fourdifferentadiabaticprocessesfromφ↑=φ↓=0toreviewoftheQAHstateinthissubsection.φ↑=φ↓=±π.Thered(blue)curvestandsforthefluxφ↑(↓)(t),respectively.Thesymbol“⊙”(“⊗”)representsincreasing(decreas-Asastartingpoint,considertheupper2×2blockoftheing)fluxes,andthearrowsshowthecurrentintoandoutoftheGaus-QSHHamiltonian(2):sianloop,inducedbythechangingflux.Chargeispumpedintheprocesseswithφa↑(t)=−φ↓(t),whilespinispumpedinthosewithh(k)=ǫ(k)I2×2+da(k)σ.(28)φ↑(t)=φ↓(t).FromQiandZhang,2008.Ifweconsideronlythesetwobands,thismodeldescribesaApplyingsimilarargumentstoprocess(b)gives∆Q↑=TRsymmetrybreakingsystem(Qietal.,2006).Aslongas∆Q↓=e/2,whichleadstoachargeonstatewith∆Q=ethereisagapbetweenthetwobands,theHallconductanceofand∆Sz=0.Processes(c)and(d)give∆Q↑=−∆Q↓=thesystemisquantized(Thoulessetal.,1982).Thequantizede/2and∆Q↑=−∆Q↓=−e/2respectively,whichyieldtheHallconductanceisdeterminedbythefirstChernnumberofspinonstateswith∆Q=0and∆Sz=±1/2.TheHamilto-theBerryphasegaugefieldintheBrillouinzone,which,forniansH(t)inthepresenceofthegaugefluxarethesameatthegenerictwo-bandmodel(28),reducestothefollowingfor-t=0andt=1,butdifferintheintermediatestagesofthemula:fouradiabaticprocesses.AssumingthatthegroundstateisZZ!e21∂dˆ∂dˆuniqueatt=0,weobtainfourfinalstatesatt=1,whichareσH=dkxdkydˆ·×,(29)theholon,chargeonandthetwospinonstates.Boththespinh4π∂kx∂ky 15whichise2/htimesthewindingnumberoftheunitvectortheHamiltonian(2),weseethatthemasstermMfortheup-dˆ(k)=d(k)/|d(k)|aroundtheunitsphere.Thed(k)vec-perblockisreplacedbyM+(GE−GH)/2,whilethatforthetordefinedinEq.(3)hasaskyrmionstructureforM/B>0lowerblockisreplacedbyM−(GE−GH)/2.Therefore,thewithwindingnumber1,whilethewindingnumberis0fortwoblocksdoacquireadifferentmass,whichmakesitpossi-M/B<0.JustasinanordinaryQHinsulator,thesystembletoreachtheQAHphase.AfterconsideringtheeffectofthewithnontrivialHallconductancee2/hhasonechiraledgeidentityterm(GE+GH)/2,theconditionfortheQAHphasestatepropagatingontheedge.FortheQSHsystemdescribedisgivenbyGEGH<0.WhenGEGH>0andGE6=GH,byEq.(2),thelower2×2blockhastheoppositeHallconduc-thetwoblocksstillacquireadifferentmass,butthesystembe-tance,sothatthetotalHallconductanceiszero,asguaranteedcomesmetallicbeforethetwoblocksdevelopanoppositeHallbyTRsymmetry.ThechiraledgestateoftheQAHanditsTRconductance.Physically,wecanalsounderstandthephysicspartnerformthehelicaledgestatesoftheQSHinsulator.fromtheedgestatepicture[Fig.9(b)].OntheboundaryofaQSHinsulatortherearecounter-propagatingedgestatescar-ryingoppositespin.Whenthespinsplittingtermincreases,oneofthetwoblocks,saythespindownblock,experiencesatopologicalphasetransitionatM=(GE−GH)/2.Thespindownedgestatespenetratedeeperintothebulkduetothede-creasinggapandeventuallydisappear,leavingonlythespinupstateboundmorestronglytotheedge.Thus,thesystemhasonlyspinupedgestatesandevolvesfromtheQSHstatetotheQAHstate[Fig.9(b)].Althoughthediscussionaboveisbasedonthespecificmodel(2),themechanismtogener-ateaQAHinsulatorfromaQSHinsulatorisgeneric.AQSHinsulatorcanalwaysevolveintoaQAHinsulatoronceaTRsymmetrybreakingperturbationisintroduced.Fortunately,inMn-dopedHgTeQWstheconditionGEGH<0isindeedsatisfied,sothattheQAHphaseex-istsinthissystemaslongastheMnspinsarepolarized.ThemicroscopicreasonfortheoppositesignofGEandGHistheoppositesignofthes-dandp-dexchangecouplingsinFIG.9Evolutionofbandstructureandedgestatesuponincreasingthissystem(Liuetal.,2008).Interestingly,inanotherfamilythespinsplitting.For(a)GE<0andGH>0,thespindownofQSHinsulators,Bi2Se3andBi2Te3thinfilms(Liuetal.,states|E1,−iandn|H1,−iinthesameblockoftheHamiltonian2010),theconditionGEGH<0isalsosatisfiedwhenmag-(2)firsttoucheachother,andthenenterthenormalregime.For(c)neticimpuritiessuchasCrorFeareintroducedintothesys-GE>0andGH>0,gapclosingoccursbetween|E1,+iandtem,butforadifferentphysicalreason.InHgTeQWs,the|H1,−i,whichbelongtodifferentblocksoftheHamiltonian,andtwobandsintheupperblockoftheHamiltonian(2)havethuswillcrosseachotherwithoutopeningagap.(b)Behaviorofthethesamedirectionofspin,butcouplewiththeimpurityspinedgestatesduringthelevelcrossing.FromLiuetal.,2008.withanoppositesignofexchangecouplingbecauseonebandoriginatesfroms-orbitalswhiletheotheroriginatesfromp-WhenTRsymmetryisbroken,thetwospinblocksarenoorbitals.InBi2Se3andBi2Te3,bothbandsoriginatefromlongerrelated,andtheirchargeHallconductancesnolongerp-orbitals,whichhavethesamesignofexchangecouplingcancelexactly.Forexample,wecanconsideradifferentmasswiththeimpurityspin,butthesignofspinintheupperblockMforthetwoblocks,whichbreaksTRsymmetry.Ifoneisopposite(Yuetal.,2010).Consequently,theconditionblockisinthetrivialinsulatorphase(M/B<0)andtheotherGEGH<0isstillsatisfied.MoredetailsonthepropertiesblockisintheQAHphase(M/B>0),thewholesystem2oftheBi2Se3andBi2Te3familyofmaterialscanbefoundbecomesaQAHstatewithHallconductance±e/h.Physi-inthenextsection,sinceasbulkmaterialstheyareboth3Dcally,thiscanberealizedbyexchangecouplingwithmagnetictopologicalinsulators.impurities.Inasystemdopedwithmagneticimpurities,thespinsplittingterminducedbythemagnetizationisgenericallywrittenasF.ExperimentalresultsGE0000GH001.QuantumwellgrowthandthebandinversiontransitionHs=,(30)00−GE0Asshownabove,thetransitionfromanormaltoanin-000−GHvertedbandstructurecoincideswiththephasetransitionfromwhereGEandGHdescribethesplittingofE1andH1bandsatrivialinsulatortotheQSHinsulator.Inordertocoverrespectively,whicharegenericallydifferent.AddingHstoboththenormalandtheinvertedbandstructureregime,HgTe 16QWsampleswithaQWwidthintherangefrom4.5nmto12.0nmweregrown(K¨onig,2007;K¨onigetal.,2008;100(a)d=40ÅK¨onigetal.,2007)bymolecularbeamepitaxy(MBE).Sam-(b)d=150ÅQWQWpleswithmobilitiesofseveral105cm2/(V·s),evenforlow40densitiesn<5×1011cm−2,wereavailablefortransport50measurements.Insuchsamples,themeanfreepathisofthe20orderofseveralmicrons.FortheinvestigationoftheQSH0effect,devicesinaHallbargeometry[Fig.12,inset]ofvari-/meV/meVE0EousdimensionswerefabricatedfromQWstructureswithwellwidthsof4.5nm,5.5nm,6.4nm,6.5nm,7.2nm,7.3nm,-508.0nmand12.0nm.-20FortheinvestigationoftheQSHeffect,sampleswithalow-100intrinsicdensityn(Vg=0)<5×1011cm−2werestudied.051015051015WhenanegativegatevoltageVgisappliedtothetopgateB/TB/Telectrodeofthedevice,theusualdecreaseinelectrondensityisobserved.InFig.10(a),measurementsoftheHallresistanceFIG.11LandauleveldispersionforquantumwellthicknessesRxyarepresentedforaHallbarwithlengthL=600µmandof(a)4.0nm,(b)15.0nm.Thequalitativebehaviorisindica-widthW=200µm.Thedecreaseofthecarrierdensityistiveforsampleswith(a)normaland(b)invertedbandstructure.reflectedinanincreaseoftheHallcoefficientwhenthegateFromK¨onigetal.,2008.voltageisloweredfrom0Vto−1V.Inthisvoltagerange,thedensitydecreaseslinearlyfrom3.5×1011cm−2to0.5×1011cm−2[Fig.10(b)].Forevenlowergatevoltages,thehowever,asignificantchangeisobservedfortheLLdisper-sion[Fig.11(b)].Duetotheinversionofelectron-likeandhole-likebands,statesnearthebottomoftheconductionbandhavepredominantlypcharacter.Consequently,theenergy30ofthelowestLLdecreaseswithincreasingmagneticfield.ABOntheotherhand,statesnearthetopofthevalenceband203V=-1Vghavepredominantlyscharacter,andthehighestLLshiftsto10higherenergieswithincreasingmagneticfield.Thisleadsto)2-2acrossingofthesetwopeculiarLLsforaspecialvalueof)V=00cmgthemagneticfield.Thisbehaviorhasbeenobservedearlier(k11xy1bytheW¨urzburggroupandcannowbedemonstratedanalyt-R(10-10nicallywithintheBHZmodel(K¨onigetal.,2008).TheexactmagneticfieldBcrossatwhichthecrossingoccursdependson-200V=-2VdQW.TheexistenceoftheLLcrossingisaclearsignatureVgthofaninvertedbandstructure,whichcorrespondstoanegative-30024680,0-0,5-1,0-1,5-2,0energygapwithM/B<0intheBHZmodel.Thecross-V(V)B(T)gingoftheLLsfromtheconductionandvalencebandscanbeobservedinexperiments[Fig.12(a)].ForgatevoltagesFIG.10(a)HallresistanceRxyforvariousgatevoltages,indicatingVg≥−1.0VandVg≤−2.0V,EFisclearlyintheconduc-thetransitionfromn-top-conductance.(b)Gate-voltagedependenttionbandandvalenceband,respectively.WhenEFisshiftedcarrierdensitydeducedfromHallmeasurements.FromK¨onigetal.,towardsthebottomoftheconductionband,i.e.Vg<−1.0V,2008.atransitionfromaQHstatewithfillingfactorν=1,i.e.R=h/e2=25.8kΩ,toaninsulatingstateisobserved.xysamplebecomesinsulating,becausetheFermienergyEFisSuchbehaviorisexpectedindependentlyofthedetailsoftheshiftedintothebulkgap.WhenalargenegativevoltageVg≤bandstructure,whenthelowestLLoftheconductionband−2Visapplied,thesamplebecomesconductingagain.ItcancrossesEFforafinitemagneticfield.WhenEFislocatedbeinferredfromthechangeinsignoftheHallcoefficientthatwithinthegap,anontrivialbehaviorcanbeobservedforde-thedeviceisp-conducting.Thus,EFhasbeenshiftedintotheviceswithaninvertedbandstructure.SincethelowestLLofvalenceband,passingthroughtheentirebulkgap.theconductionbandlowersitsenergywithincreasingmag-ThepeculiarbandstructureofHgTeQWsgivesrisetoneticfield,itwillcrossEFforacertainmagneticfield.Subse-auniqueLLdispersion.Foranormalbandstructure,i.e.,quently,oneoccupiedLLisbelowEF,givingrisetotheusualdQWdc,cross.Uponcrossing,their”character”isexchanged,i.e.the 17fortheexistenceoftheQSHstate.Incontrast,trivialinsu-latingbehaviorisobtainedfordeviceswithdQWdc2FIG.12(a)HallresistanceRxyofa(L×W)=(600×200)µmQWstructurewith6.5nmwellwidthfordifferentcarrierconcen-trationsobtainedfordifferentgatevoltagesVgintherangefrom−1Vto−2V.FordecreasingVg,then-typecarrierconcentrationdecreasesandatransitiontop-typeconductionisobserved,passingthroughaninsulatingregimebetween−1.4Vand−1.9Vatzerofield.(b)Landaulevelfanchartofa6.5nmquantumwellobtainedfromaneight-bandk·pcalculation.BlackdashedlinesindicatethepositionoftheFermienergy,EF,forgatevoltages−1.0Vand−2.0V.RedandgreendashedlinesindicatethepositionofEFfortheredandgreenHallresistancetracesin(a).ThecrossingpointsofEFwiththerespectiveLandaulevelsaremarkedbyarrowsofthesamecolor.FromK¨onigetal.,2007.levelfromthevalencebandturnsintoaconductionbandLLFIG.13Longitudinalresistanceofa4.5nmQW[dashed(black)]andviceversa.ThelowestLLoftheconductionbandnowanda8.0nmQW[solid(red)]asafunctionofgatevoltage.risesinenergyforlargermagneticfields.Consequently,itFromK¨onigetal.,2008.willcrosstheEFforacertainmagneticfield.SinceEFwillbelocatedwithinthefundamentalgapagainafterwards,theandaninvertedbandstructure,however,theresistancedoessamplewillbecomeinsulatingagain.Suchareentrantn-typenotexceed100kΩ.Thisbehaviorisreproducedforvari-QHstateisshowninFig.12(a)forVg=−1.4V(greentrace).ousHallbarswithaQWwidthintherangefrom4.5nmtoForlowergatevoltages,acorrespondingbehaviorisobserved12.0nm.Whiledeviceswithanormalbandstructure,i.e.forap-typeQHstate(e.g.redtraceforVg=−1.8V).AsdQWλc,ofBi2Se3attheΓpointversusanartificiallyrescaledatomicSOCtheorderofthesetwoenergylevelsisreversed.Toillustrateλ(Bi)=xλ0(Bi)=1.25x[eV],λ(Se)=xλ0(Se)=0.22x[eV]thisinversionprocessexplicitly,theenergylevels|P1+iand(seetext).Alevelcrossingoccursbetweenthesetwostatesatz|P2−ihavebeencalculated(Zhangetal.,2009)foramodelx=xc≃0.6.AdaptedfromZhangetal.,2009.zHamiltonianofBi2Se3withartificiallyrescaledatomicSOCparametersλ(Bi)=xλ0(Bi),λ(Se)=xλ0(Se),asshowninants(Winkler,2003)atafinitewavevectork.TheimportantFig.19(b).Hereλ0(Bi)=1.25eVandλ0(Se)=0.22eVsymmetriesofthesystemareTRsymmetryT,inversionsym-aretheactualvaluesoftheSOCstrengthforBiandSeatoms,metryI,andthree-foldrotationsymmetryC3aroungthezrespectively(WittelandManne,1974).FromFig.19(b),oneaxis.Inthebasis{|P1+,↑i,|P2−,↑i,|P1+,↓i,|P2−,↓i},canclearlyseethatalevelcrossingoccursbetween|P1+izzzzz−therepresentationofthesesymmetryoperationsisgivenand|P2ziwhentheSOCstrengthisabout60%ofitsactualyvalue.Sincethesetwolevelshaveoppositeparity,theinver-byT=iσK⊗
I2×2,I=I2×2⊗τ3andC3=expiπσz⊗I,whereIisthen×nidentityma-sionbetweenthemdrivesthesystemintoatopologicalinsu-32×2n×ntrix,Kisthecomplexconjugationoperator,andσx,y,zandlatorphase,similartothecaseofHgTeQWs(Bernevigetal.,τx,y,zdenotethePaulimatricesinthespinandorbitalspace,2006).Therefore,themechanismfortheoccurrenceofa3Drespectively.Byrequiringthesethreesymmetriesandkeepingtopologicalinsulatingphaseinthissystemiscloselyanalo-onlytermsuptoquadraticorderink,weobtainthefollowinggoustothemechanismforthe2DQSHeffect(2Dtopolog-genericformoftheeffectiveHamiltonian:icalinsulator)inHgTe(Bernevigetal.,2006).Morepre-cisely,todeterminewhetherornotaninversion-symmetricH(k)=ǫ0(k)I4×4+crystalisatopologicalinsulator,wemusthavefullknowl-edgeofthestatesatalloftheeightTRinvariantmomentaM(k)A1kz0A2k−(TRIM)(FuandKane,2007).Thesystemisa(strong)topo-A1kz−M(k)A2k−0,(31)logicalinsulatorifandonlyifthebandinversionbetween0A2k+M(k)−A1kzstateswithoppositeparityoccursatoddnumberofTRIM.A2k+0−A1kz−M(k)TheparityoftheBlochstatesatallTRIMhavebeenstudiedwithk=k±ik,ǫ(k)=C+Dk2+Dk2andM(k)=byabinitiomethodsforthefourmaterialsBi2Se3,Bi2Te3,±xy01z2⊥M−Bk2−Bk2.TheparametersintheeffectivemodelcanSb2Se3,andSb2Te3(Zhangetal.,2009).Comparingthe1z2⊥BlochstateswithandwithoutSOC,oneconcludethatSb2Se3bedeterminedbyfittingtheenergyspectrumoftheeffectiveisatrivialinsulator,whiletheotherthreearetopologicalinsu-Hamiltoniantothatofabinitiocalculations(Liuetal.,2010;lators.Forthethreetopologicalinsulators,thebandinversionZhangetal.,2009,2010).ThefittingleadstotheparametersonlyoccursattheΓpoint.displayedinTableII(Liuetal.,2010).Exceptfortheidentitytermǫ0(k),theHamiltonian(31)isSincethetopologicalnatureisdeterminedbythephysicssimilartothe3DDiracmodelwithuniaxialanisotropyalongneartheΓpoint,itispossibletowritedownasimpleeffectivethezdirection,butwiththecrucialdifferencethatthemassHamiltoniantocharacterizethelow-energy,long-wavelengthtermisk-dependent.FromthefactthatM,B1,B2>0propertiesofthesystem.Startingfromthefourlow-lyingwecanseethattheorderofthebands|T1+,↑(↓)iandzstates|P1+,↑(↓)iand|P2−,↑(↓)iattheΓpoint,such|T2−,↑(↓)iisinvertedaroundk=0comparedwithlargezzzaHamiltoniancanbeconstructedbythetheoryofinvari-k,whichcorrectlycharacterizesthetopologicallynontrivial 22ConsiderthemodelHamiltonian(31)onthehalf-spacez>TABLEIITheparametersinthemodelHamilto-0.Inthesamewayasinthe2Dcase,wecandividethemodelnian(31)obtainedfromfittingtoabinitiocalculation.Adaptedfrom(Liuetal.,2010).Hamiltonianintotwoparts,Bi2Se3Bi2Te3Sb2Te3Hˆ=H˜0+H˜1,(32)A1(eV·A)˚2.260.300.84M˜(kz)A1kz00A2(eV·A)˚3.332.873.40Ak−M˜(k)00H˜=˜ǫ(k)+1zz,C(eV)-0.0083-0.180.0010z00M˜(kz)−A1kz2D1(eV·A˚)5.746.55-12.3900−A1kz−M˜(kz)D2(eV·A˚2)30.449.68-10.78−Bk200AkM(eV)0.280.300.222⊥2−2B1(eV·A˚2)6.862.7919.64H˜=Dk2+0B2k⊥A2k−0,(33)12⊥2B2(eV·A˚2)44.557.3848.510A2k+−B2k⊥0Ak00Bk22+2⊥withǫ˜(k)=C+Dk2andM˜(k)=M−Bk2.H˜natureofthesystem.Inaddition,theDiracmassM,i.e.thez1zz1z0inEq.(8)andEq.(32)areidentical,withtheparametersbulkinsulatinggap,is∼0.3eV,whichallowsthepossibil-A,B,C,D,MinEq.(8)replacedbyA1,B1,C,D1,Minityofhavingaroom-temperaturetopologicalinsulator.SuchEq.(32).Therefore,thesurfacestateatkx=ky=0isdeter-aneffectivemodelcanbeusedforfurthertheoreticalstudyofminedbythesameequationasthatfortheQSHedgestates.AtheBi2Se3system,aslongaslow-energypropertiesarecon-surfacestatesolutionexistsforM/B1>0.Inthesamewaycerned.asinthe2Dcase,thesurfacestatehasahelicitydeterminedCorrectionstotheeffectiveHamiltonian(31)thatareof3bythesignofA1/B1.(Hereandbelowwealwaysconsiderhigherorderinkcanalsobeconsidered.Tocubic(k)thecasewithB1B2>0,A1A2>0.)order,somenewtermscanbreakthecontinuousrotationInanalogytothe2DQSHcase,thesurfaceeffectivemodelsymmetryaroundthezaxistoadiscretethree-foldrotationcanbeobtainedbyprojectingthebulkHamiltonianontosymmetryC3.Correspondingly,theFermisurfaceofthethesurfacestates.Totheleadingorderinkx,ky,theef-surfacestateacquiresahexagonalshape(Fu,2009),whichfectivesurfaceHamiltonianHsurfhasthefollowingmatrixleadstoimportantconsequencesforexperimentsontopo-form(Liuetal.,2010;Zhangetal.,2009):logicalinsulatorssuchassurfacestatequasiparticleinterfer-ence(Alpichshevetal.,2010;Leeetal.,2009;Zhangetal.,H(k,k)=C+A(σxk−σyk).(34)surfxy2yx2009;Zhouetal.,2009).Amodifiedversionoftheeffective3model(31)takingintoaccountcorrectionsuptok3hasbeenHigherordertermssuchasktermsbreaktheaxialsymme-obtainedforthethreetopologicalinsulatorsBi2Se3,Bi2Te3,tryaroundthezaxisdowntoathree-foldrotationsymmetry,andSb2Te3basedonabinitiocalculations(Liuetal.,2010).whichhasbeenstudiedintheliterature(Fu,2009;Liuetal.,Inthissamework(Liuetal.,2010),aneight-bandmodelis2010).ForA2=4.1eV·A,thevelocityofthesurfacestatesis˚givenbyv=A/~≃6.2×105m/s,whichagreesreasonablyalsoproposedforamorequantitativedescriptionofthisfam-2withabinitioresults[Fig.20]v≃5.0×105m/s.ilyoftopologicalinsulators.Tounderstandthephysicalpropertiesofthesurfacestates,weneedtoanalyzetheformofthespinoperatorsinthisB.SurfacestateswithasingleDiracconesystem.Byusingthewavefunctionfromabinitiocalcu-lationsandprojectingthespinoperatorsontothesubspaceTheexistenceoftopologicalsurfacestatesisoneofthespannedbythefourbasisstates,weobtainthespinoperatorsmostimportantpropertiesoftopologicalinsulators.Thesur-forourmodelHamiltonian,withmatrixelementsbetweensur-facestatescanbedirectlyextractedfromabinitiocalcula-facestatesgivenbyhΨ|S|Ψi=Sσαβ,hΨ|S|Ψi=αxβx0xαyβtionsbyconstructingmaximallylocalizedWannierfunctionsSσαβandhΨ|S|Ψi=Sσαβ,withSsomepos-y0yαzβz0zx(y,z)0andcalculatingthelocaldensityofstatesonanopenbound-itiveconstants.Therefore,weseethatthePauliσmatrixary(Zhangetal.,2009).TheresultfortheBi2Se3familyofinthemodelHamiltonian(34)isproportionaltothephysi-materialsisshowninFig.20(a)-(d),whereonecanclearlycalspin.Asdiscussedabove,thespindirectionisdeterminedseethesingleDirac-conesurfacestateforthethreetopo-bythesignoftheparameterA1/B1,whichdependsonmate-logicallynontrivialmaterials.However,toobtainabetterrialpropertiessuchastheatomicSOC.IntheBi2Se3familyunderstandingofthephysicaloriginoftopologicalsurfaceofmaterials,theupperDiracconehasaleft-handedhelicitystates,itishelpfultoshowhowthesurfacestatesemergefromwhenlookingfromabovethesurface[Fig.20(e),(f)].theeffectivemodel(31)(Linderetal.,2009;Liuetal.,2010;Fromthediscussionabove,weseethatthesurfacestateLuetal.,2010;Zhangetal.,2009).Thesurfacestatescanbeisdescribedbya2DmasslessDiracHamiltonian(34).An-obtainedinasimilarwayastheedgestatesoftheBHZmodelotherwell-knownsystemwithasimilarpropertyisgraphene,(Sec.II.B).asinglesheetofgraphite(CastroNetoetal.,2009).However, 23wardlygeneralizedtotheinterfacestatesbetweentwoinsu-lators(Fradkinetal.,1986;VolkovandPankratov,1985).Inthesepioneeringworks,theinterfacestatesbetweenPbTeandSnTewereinvestigated.TheinterfacestatesconsistoffourDiraccones.Therefore,theyaretopologicallytrivialandnotgenerallystableunderTRinvariantperturbations.Thesurfacestatesoftopologicalinsulatorsarealsosimilartothedomainwallfermionsoflatticegaugetheory(Kaplan,1992).Infact,domainwallfermionsarepreciselyintroducedtoavoidthefermiondoublingproblemonthelattice,whichissimilartotheconceptofasingleDiracconeonthesurfaceofatopolog-icalinsulator.ThehelicalspintexturedescribedbythesingleDiracconeequation(34)leadstoageneralrelationbetweenchargecur-rentdensityj(x)andspindensityS(x)onthesurfaceofthetopologicalinsulator(Raghuetal.,2010):j(x)=v[ψ†(x)σψ(x)×zˆ]=vS(x)×zˆ.(35)Inparticular,theplasmonmodeonthesurfacegenerallycar-riesspin(BurkovandHawthorn,2010;Raghuetal.,2010).C.CrossoverfromthreedimensionstotwodimensionsFromthediscussionabove,onecanseethatthemodelsdescribing2Dand2Dtopologicalinsulatorsarequitesimi-lar.BothsystemsaredescribedbylatticeDirac-typeHamil-tonians.Inparticular,wheninversionsymmetryispresent,FIG.20(a)-(d)Energyandmomentumdependenceofthelocalden-thetopologicallynontrivialphaseinbothmodelsischarac-sityofstatesfortheBi2Se3familyofmaterialsonthe[111]surface.terizedbyabandinversionbetweentwostatesofoppositeAwarmercolorrepresentsahigherlocaldensityofstates.Redre-parity.Therefore,itisnaturaltostudytherelationbetweengionsindicatebulkenergybandsandblueregionsindicateabulkthesetwotopologicalstatesofmatter.Onenaturalquestionenergygap.ThesurfacestatescanbeclearlyseenaroundΓpointiswhetherathinfilmof3Dtopologicalinsulator,viewedasasredlinesdispersinginsidethebulkgap.(e)Spinpolarizationofthesurfacestatesonthetopsurface,wherethezdirectionisthesur-a2Dsystem,isatrivialinsulatororaQSHinsulator.Be-facenormal,pointingoutwards.AdaptedfromZhangetal.,2009sidestheoreticalinterest,thisproblemisalsorelevanttoex-andLiuetal.,2010.periments,especiallyintheBi2Se3familyofmaterials.In-deed,thesematerialsarelayeredandcanbeeasilygrownasthinfilmseitherbyMBE(Lietal.,2010,2009;Zhangetal.,thereisakeydifferencebetweenthesurfacestatetheoryfor2009),catalyst-freevapor-solidgrowth(Kongetal.,2010),3Dtopologicalinsulatorsandgrapheneorany2DDiracsys-orbymechanicalexfoliation(Hongetal.,2010;Shahiletal.,tem,whichisthenumberofDiraccones.Graphenehasfour2010;Teweldebrhanetal.,2010).SeveraltheoreticalworksDiracconesatlowenergies,duetospinandvalleydegener-studiedthinfilmsoftheBi2Se3familyoftopologicalinsu-acy.ThevalleydegeneracyoccursbecausetheDiracconeslators(Linderetal.,2009;Liuetal.,2010;Luetal.,2010).arenotinthevicinityofk=0butrathernearthetwoBril-Interestingly,thinfilmsofproperthicknessesarepredictedtolouinzonecornersKandK¯.Thisisgenericforapurely2DformaQSHinsulator(Liuetal.,2010;Luetal.,2010),whichsystem:onlyanevennumberofDiracconescanexistinaTRmayconstituteanapproachforsimplerrealizationsofthe2Dinvariantsystem.Inotherwords,asingle2DDiracconewith-QSHeffect.outTRsymmetrybreakingcanonlyexistonthesurfaceofaSuchacrossoverfrom3Dto2Dtopologicalinsulatorscantopologicalinsulator,whichisalsoanalternativewaytoun-bestudiedfromtwopointsofview,eitherfromthebulkstatesderstanditstopologicalrobustness.AslongasTRsymmetryofthe3Dtopologicalinsulatororfromthesurfacestates.ispreserved,thesurfacestatecannotbegappedoutbecauseWefirstconsiderthebulkstates.Athinfilmof3Dtopo-nopurely2DsystemcanprovideasingleDiraccone.Suchalogicalinsulatorisdescribedbyrestrictingthebulkmodelsurfacestateisa“holographicmetal”whichis2Dbutdeter-(31)toaQWwiththicknessd,outsidewhichthereisanin-minedbythe3Dbulktopologicalproperty.finitebarrierdescribingthevacuum.Toestablishthecon-Inthissectionwediscussedthesurfacestatesofaninsula-nectionbetweenthe2DBHZmodel(2)andthe3Dtopo-torsurroundedbyvacuum.Thisformalismcanbestraightfor-logicalinsulatormodel(31),westartfromthespecialcase 24A1=0andconsiderafiniteA1lateron.ForA1=0andTherefore,thecompleteeffectiveHamiltonianisgivenbykx=ky=0,theHamiltonian(31)becomesdiagonalandtheSchr¨odingerequationfortheinfiniteQWcanbeeasilysolved.0ik−00qTheHamiltonianeigenstatesaresimplygivenby|En(Hn)i=H(k,k)=A−ik+000.
surfxy22sinnπz+nπ|Λi,with|Λi=|P1+z,↑(↓)iforelec-000−ik−dd2−00ik+0tronsubbandsand|Λi=|P2z,↑(↓)iforholesubbands.ThecorrespondingenergyspectrumisEe(n)=C+M+(D1−Whenaslaboffinitethicknessisconsidered,thetwosurfacenπ
2nπ
2B1)dandEh(n)=C−M+(D1+B1)d,respec-statesstartoverlapping,suchthatoff-diagonaltermsareintro-tively.WeassumeM<0andB1<0sothatthesystemducedintheeffectiveHamiltonian.AneffectiveHamiltonianstaysintheinvertedregime.TheenergyspectrumisshowninconsistentwithinversionandTRsymmetryandincorporatingFig.21(a).Whenthewidthdissmallenough,electronsub-inter-surfacetunnelingisgivenbybandsEnhaveahigherenergythantheholesubbandsHnduetoquantumconfinementeffects.Becausethebulkbands0ik−M2D0areinvertedattheΓpoint(M<0),theenergyoftheelec-−ik+00M2DHsurf(kx,ky)=A2,(36)tronsubbandswilldecreasewithincreasingdtowardstheirM2D00−ik−bulkvalueM<0,whiletheenergyoftheholesubbands0M2Dik+0willincreasetowards−M>0.Therefore,theremustexistacrossingpointbetweentheelectronandholesubbands.whereM2DisaTRinvariantmasstermduetointer-surfacetunneling,whichgenerallydependsonthein-planemomen-tum.Equation(36)isunitarilyequivalenttotheBHZHamil-WhenafiniteA1isturnedon,theelectronandholebandstonian(2)forHgTeQWs.WhethertheHamiltoniancorre-arehybridizedsothatsomeofthecrossingsbetweentheQWspondstoatrivialorQSHinsulatorcannotbedeterminedlevelsareavoided.However,asshowninFig.21(b),somewithoutstudyingthebehaviorofthismodelatlargemomenta.levelcrossingscannotbelifted,whichisaconsequenceofin-Indeed,wearemissingaregularizationtermwhichwouldversionsymmetry.Whenthebandindexnisincreased,theplaytheroleofthequadratictermBk2intheBHZmodelparityofthewavefunctionsalternatesforbothelectronand(Sec.II.A).However,thetransitionsbetweentrivialandnon-holesubbands.Moreover,theatomicorbitalsformingelec-trivialphasesareaccompaniedbyasignchangeinM2D,in-tronandholebandsare|P1+,↑(↓)iand|P2−,↑(↓)ire-zzdependentlyoftheregularizationschemeathighmomenta.spectively,whichhaveoppositeparity.Consequently,|EniUponvariationofd,thesignoftheinter-surfacecouplingand|Hniwiththesameindexnhaveoppositeparity,sothatM2Doscillatesbecausethesurfacestatewavefunctionsos-theircrossingcannotbeavoidedbytheA1term.Whenacillate[Fig.21(b)].Therefore,wereachthesameconclusionsfinitekx,kyisconsidered,eachlevelbecomesaQWsub-asinthebulkapproach.band.ThebottomofthelowestconductionbandandthetopTheresultsaboveobtainedfromcalculationsusinganef-+−ofthehighestvalencebandareindicatedbyS1andS2infectivemodelarealsoconfirmedbyfirst-principlecalcula-Fig.21(b).Sincethesetwobandshaveoppositeparity,eachtions.Theparityeigenvaluesofoccupiedbandshavebeenlevelcrossingbetweenthemisatopologicalphasetransitioncalculatedasafunctionofthethicknessofthe3Dtopologi-betweentrivialandQSHinsulatorphases(Bernevigetal.,calinsulatorfilm(Liuetal.,2010),fromwhichthetopologi-2006;FuandKane,2007).Sincethesystemmustbetriv-calnatureofthefilmcanbeinferred.Theresultisshowninialinthelimitd→0,weknowthatthefirstQSHinsulatorFig.21(d),whichconfirmstheoscillationsfoundintheeffec-phaseoccursbetweenthefirstandsecondlevelcrossing.Intivemodel.ThefirstnontrivialphaseappearsatathicknessoftheA1→0limit,thecrossingpositionsaregivenbythecrit-qthreequintuplelayers,i.e.about3nmforBi2Se3.icalwellthicknessesd=nπB1.Inprinciple,thereiscn|M|aninfinitenumberofQSHphasesbetweendc,2n−1anddc,2n.+D.ElectromagneticpropertiesHowever,asseeninFig.21(b),thegapbetween|S1iand−|S2idecaysquicklyforlarged.Inthe3Dlimitd→∞,theInprevioussubsections,wehavereviewedbulkandsur-twostatesbecomedegenerateandactuallyformthetopandfacepropertiesof3Dtopologicalinsulators,aswellastheirbottomsurfacestatesofthebulkcrystal[Fig.21(c)].relationto2Dtopologicalinsulators(QSHinsulators),basedonamicroscopicmodel.FromtheeffectivemodelofsurfaceThisrelationbetweentheQWvalenceandconductionstates,onecanunderstandtheirrobustnessprotectedbyTRbandsandthesurfacestatesinthed→∞limitsuggestsansymmetry.However,similarlytothequantizedHallresponsealternativewaytounderstandthecrossoverfrom3Dto2D,inQHsystems,thetopologicalstructureintopologicalinsu-i.e.fromthesurfacestates.Inthe3Dlimitthetwosurfaceslatorsshouldnotonlyleadtorobustgaplesssurfacestates,aredecoupledandaretheonlylow-energystates.Thetopsur-butalsotounique,quantizedelectromagneticresponsecoef-faceisdescribedbytheeffectiveHamiltonian(34)whiletheficients.Thequantizedelectromagneticresponseof3Dtopo-bottomsurfaceisobtainedfromthetopsurfacebyinversion.logicalinsulatorsturnsouttobeatopologicalmagnetoelectric 25(a)(b)couplingbetweentheconescanbeintroduced,whichleadsto0.5E+E-0.5S+thegappedTRinvariantHamiltonian121!00A(σxk−σyk)−imσz′2yxE(eV)E(eV)Hsurf(k)=zxy.H-H+S-imσA2(σky−σkx)122-0.5-0.5dc1dc2dc1dc21234512345Fromsuchadifferencebetweenanevenandanoddnumberd(nm)d(nm)(c)(d)ofDiraccones,oneseesthatthestabilityofthesurfacetheory(34)isprotectedbyaZ2topologicalinvariant.0.042AlthoughthesurfacestatewithasingleDiracconedoesnotremaingaplesswhenaTRbreakingmasstermmσzis(z)|z0.02Y|Yadded,animportantphysicalpropertyisinducedbysuchamassterm:ahalf-integerquantizedHallconductance.As0discussedinSec.II.E,theHallconductanceofageneric-10-50510z(nm)two-bandHamiltonianh(k)=d(k)σaisdeterminedbyaEq.(29),whichisthewindingnumberoftheunitvectordˆ(k)=d(k)/|d(k)|ontheBrillouinzone.TheperturbedFIG.21Energylevelsversusquantumwellthicknessfor(a)A1=surfacestateHamiltonian0eV·A,(b)˚A1=1.1eV·A.Otherparametersaretakenfrom˚Zhangetal.,2009.ShadedregionsindicatetheQSHregime.Thexyzbluedashedlinein(b)showshowthecrossingbetween|E1(H1)iHsurf(k)=A2σky−A2σkx+mzσ,(37)and|H2(E2)ievolvesintoananti-crossingwhenA16=0.(c)Probabilitydensityinthestate|S+i(samefor|S−i)forA1=correspondstoavectord(k)=(A2ky,−A2kx,mz).At121.1eV·Aand˚d=20nm.(d)Bandgapandtotalparityfromabk=0,theunitvectordˆ(k)=(0,0,mz/|mz|)pointsto-initiocalculationsonBi2Se3,plottedasafunctionofthenumberofwardsthenorth(south)poleoftheunitsphereformz>0quintuplelayers.FromLiuetal.,2010.(mz<0).For|k|≫|mz|/A2,theunitvectordˆ(k)≃A2(ky,−kx,0)/|k|almostliesintheequatorialplaneoftheunitsphere.Fromsucha“meron”configurationoneseesthateffect(TME)(Qietal.,2008,2009),whichoccurswhenTRdˆ(k)covershalfoftheunitsphere,whichleadstoawindingsymmetryisbrokenonthesurface,butnotinthebulk.Thenumber±1/2andcorrespondstoaHallconductanceTMEeffectisagenericpropertyof3Dtopologicalinsula-tors,whichcanbeobtainedtheoreticallyfromgenericmodelsme2zσH=.(38)andfromaneffectivefieldtheoryapproach(Essinetal.,2009;|mz|2hFuandKane,2007;Qietal.,2008),independentlyofmicro-scopicdetails.However,inordertodevelopaphysicalintu-Fromthisformula,itcanbeseenthattheHallconductanceitionfortheTMEeffect,inthesectionwereviewthiseffectremainsfiniteeveninthelimitmz→0,andhasajumpatanditsphysicalconsequencesbasedonthesimplestsurfacemz=0.AsapropertyofthemassiveDiracmodel,suchaeffectivemodel,andpostponeadiscussionintheframeworkhalfHallconductancehasbeenstudiedalongtimeagoinhighofageneraleffectivetheorytoSec.IV.Weshallalsodiscussenergyphysics.Inthatcontext,theeffectistermedthe“par-variousexperimentalmanifestationsoftheTMEeffect.ityanomaly”(Redlich,1984;Semenoff,1984),becausethemasslesstheorypreservesparity(andTR)butaninfinitesimalmasstermnecessarilybreaksthesesymmetries.Theanalysisaboveonlyappliesifthecontinuumeffec-1.HalfquantumHalleffectonthesurfacetivemodel(34)applies,i.e.ifthecharacteristicmomentumWestartbyanalyzinggenericperturbationstothe|mz|/A2ismuchsmallerthanthesizeoftheBrillouinzone2π/awithathelatticeconstant.SincedeviationsfromthiseffectivesurfacestateHamiltonian(34).TheonlyDirac-typeeffectivemodelatlargemomentaisnotincludedinmomentum-independentperturbationonecanaddisH1=PatheabovecalculationoftheHallconductance(FuandKane,a=x,y,zmaσ,andtheperturbedHamiltonianhastheq2007;Lee,2009),onecannotunambiguouslypredicttheHallspectrumE=±(Ak+m)2+(Ak−m)2+m2.k2yx2xyzconductanceofthesurface.Infact,iftheeffectivetheoryde-Thus,theonlyparameterthatcanopenagapanddestabi-scribesa2Dsystemratherthanthesurfaceofa3Dsystem,lizethesurfacestatesismz,andwewillonlyconsiderthisadditionalcontributionsfromlarge-momentumcorrectionstoperturbationinthefollowing.Themasstermmσzisoddtheeffectivemodelarenecessary,sincetheHallcondutancezunderTR,asexpectedfromthetopologicalstabilityofsur-ofanygapped2DbandinsulatormustbequantizedinintegerfacestatesprotectedbyTRsymmetry.Bycomparison,iftheunitsofe2/h(Thoulessetal.,1982).Forexample,theQAHsurfacestatesconsistofevennumberofDiraccones,onecaninsulator[Eq.(28)]withmasstermM→0isalsodescribedcheckthataTRinvariantmasstermisindeedpossible.Forbythesameeffectivetheoryas(37),buthasHallconductanceexample,iftherearetwoidenticalDiraccones,animaginary0or1ratherthan±1/2(Fradkinetal.,1986). 26Interestingly,thesurfaceofa3DtopologicalinsulatorisOnthecontrary,withTRsymmetrybreakingdisorder,differentfromall2Dinsulators,inthesensethatsuchcontri-thesystembelongstotheunitaryclass,whichexhibitslo-butionsfromlargemomentavanishduetotherequirementofcalizationforarbitrarilyweakdisorderstrength.WhileTRsymmetry(Qietal.,2008).Thisfactisdiscussedmorethelongitudinalresistivityflowstoinfinityduetolocal-rigorouslyinSec.IVbasedonthegeneraleffectivefieldthe-ization,theHallconductivityflowstothequantizedvalueory.Herewepresentanargumentbasedonthebulktosurface±e2/2h(Nomuraetal.,2008).Therefore,thesystementersrelationship.Tounderstandthis,considerthejumpinHallahalfQHphaseonceaninfinitesimalTRsymmetrybreak-conductanceatmz=0.AlthoughdeviationsfromtheDiracingperturbationisintroduced,independentlyofthedetailedeffectivemodelatlargemomentamayleadtocorrectionstoformoftheTRbreakingperturbation.Physically,TRbreak-theHallconductanceforagivenmz,thechangeinHallcon-ingdisorderisinducedbymagneticimpurities,thespinofm2ductance∆σ=σ(m→0+)−σ(m→0−)=zewhichnotonlycontributesarandomTRbreakingfield,butHHzHz|mz|hisindependentofthelarge-momentumcontributions.Indeed,alsohasitsowndynamics.Forexample,thesimplestex-theeffectofthemasstermmσzonthelarge-momentumsec-changeinteractionbetweenimpurityspinandsurfacestatezPcanbewrittenasH=JS·ψ†σψ(R)withSthetorofthetheoryisnegligibleaslongasmz→0.Therefore,intiiiiiimpurityspin,ψ†σψ(R)thespindensityofsurfaceelec-anycontributionstoσHfromlargemomentashouldbecon-itinuousfunctionsofmz,andthuscannotaffectthevalueoftronsattheimpuritypositionRi,andJitheexchangecou-thediscontinuity∆σH.Ontheotherhand,sincethesurfacepling.Tounderstandthephysicalpropertiesofthetopologi-theorywithmz=0isTRinvariant,TRtransformsthesystemcalinsulatorsurfaceinthepresenceofmagneticimpurities,itwithmassmztothatwithmass−mz.Consequently,fromTRisinstructivetostudytheinteractionbetweenimpurityspinssymmetrywehavemediatedbythesurfaceelectrons(Liuetal.,2009).Asina+−usualFermiliquid,ifthesurfacestatehasafiniteFermiwaveσH(mz→0)=−σH(mz→0).vectorkF,aRuderman-Kittel-Kasuya-Yosida(RKKY)inter-m2Togetherwiththecondition∆σ=ze,weseethattheactionbetweentheimpurityspinsisintroduced,thesignofH|mz|hhalfHallconductancegivenbyEq.(38)isrobust,andthecon-whichoscillateswithwavelength∝1/2kF(Liuetal.,2009;tributionfromlarge-momentumcorrectionsmustvanish.ByYeetal.,2010).IftheFermilevelisclosetotheDiracpoint,comparison,ina2DQAHHamiltoniandiscussedinSec.II.E,i.e.kF→0,thesignoftheRKKYinteractiondoesnoti.e.theupper2×2blockofEq.(2),theHamiltonianwithoscillatebutisuniform.ThesignoftheresultinguniformmassMisnottheTRconjugateofthatwithmass−M,andspin-spininteractionisdeterminedbythecouplingtothesur-theaboveargumentdoesnotapply.Therefore,thehalfHallfaceelectrons,whichturnsouttobeferromagnetic.Phys-conductanceisauniquepropertyofthesurfacestatesof3Dically,theinteractionisferromagneticratherthanantiferro-topologicalinsulatorswhichisdeterminedbythebulktopol-magnetic,becauseauniformspinpolarizationcanmaximizeogy.Thispropertydistinguishesthesurfacestatesof3Dtopo-thegapopenedonthesurface,whichisenergeticallyfavor-logicalinsulatorsfromallpure2Dsystems,ortopologicallyable.Duetothisferromagneticspin-spininteraction,thesys-trivialsurfacestates.temcanorderferromagneticallywhenthechemicalpotentialTheanalysisabovehasonlyconsideredtranslationallyin-isneartheDiracpoint(Liuetal.,2009).Thismechanismisvariantperturbationstothesurfacestates,buttheconclu-ofgreatpracticalimportance,becauseitprovidesawaytosionsremainrobustwhendisorderisconsidered.A2DgenerateasurfaceTRsymmetrybreakingfieldbycoatingthemetalwithoutSOCbelongstotheorthogonalorunitarysurfacewithmagneticimpuritiesandtuningthechemicalpo-symmetryclassesofrandomHamiltonians,theeigenfunc-tentialneartheDiracpoint(Chaetal.,2010;ChenandWan,tionsofwhicharealwayslocalizedwhenrandomdisorder2010;Fengetal.,2009;TranandKim,2010;Zitko,2010).isintroduced.ThiseffectisknownasAndersonlocaliza-tion(Abrahamsetal.,1979).Andersonlocalizationisaquan-tuminterferenceeffectinducedbyconstructiveinterference2.Topologicalmagnetoelectriceffectbetweendifferentbackscatteringpaths.Bycomparison,asys-temwithTRinvarianceandSOCbelongstothesymplec-Asdiscussedabove,thesurfacehalfQHeffectisauniqueticclass,wheretheconstructiveinterferencebecomesde-propertyofaTRsymmetrybreakingsurface,andisdeter-structive.Inthatcase,thesystemhasametallicphaseatminedbythebulktopology,independentlyofdetailsoftheweakdisorder,whichturnsintoaninsulatorphasebygo-surfaceTRsymmetrybreakingperturbation.Akeydiffer-ingthroughametal-insulatortransitionatacertaindisor-encebetweenthesurfacehalfQHeffectandtheusualinte-derstrength(EversandMirlin,2008;Hikamietal.,1980).gerQHeffectisthattheformercannotbemeasuredbyadcNaively,onewouldexpectthesurfacestatewithnonmagnetictransportexperiment.AnintegerQHsystemhaschiraledgedisordertobeinthesymplecticclass.However,Nomura,stateswhichcontributetothequantizedHallcurrentwhilebe-Koshino,andRyushowedthatthesurfacestateismetallicingconnectedtoleads.However,asonecaneasilyconvinceevenforanarbitraryimpuritystrength(Nomuraetal.,2007),oneself,itisasimplemathematicalfactthatthesurfaceofawhichisconsistentwiththetopologicalrobustnessofthesur-finitesampleof3Dtopologicalinsulatorisalwaysaclosedfacestate.manifoldwithoutanedge.Ifthewholesurfaceofatopolog- 27icalinsulatorsampleisgappedbymagneticimpurities,thereM(a)(b)tarenoedgestatestocarryadctransportcurrent.Ifthemag-neticimpuritiesformaferromagneticphaseandthereisado-mainwallinthemagneticmoment,theHallconductancehasajumpatthedomainwallduetotheformula(38).Inthis2case,thejumpofHallconductanceise/hacrossthedomainFMwall,sothatachiralgaplessedgestatepropagatesalongtheTIdomainwall[Fig.22(a)](Qietal.,2008).ThismechanismprovidesanotherroutetowardstheQAHeffectwithoutanyexternalmagneticfieldandtheassociatedLLs.ThisisveryFIG.22(a)Ferromagneticlayeronthesurfaceoftopologicalinsu-muchaliketheboundarybetweentwoordinaryQHstateswith22latorwithamagneticdomainwall,alongwhichachiraledgestateHallconductancene/hand(n+1)e/h.Thewavefunc-propagates.(b)RelationbetweensurfacehalfQHeffectandbulktionforanedgestatealongastraightdomainwallcanalsobetopologicalmagnetoelectriceffect.AmagnetizationisinducedbysolvedforanalyticallyfollowingthesameprocedureasthatanelectricfieldduetothesurfaceHallcurrent.FromQietal.,2008.usedinSec.II.B.Interestingly,ifoneattachesvoltageandcurrentleadstothedomainwallinthesamewayasforanordinaryHallbar,oneshouldobserveaHallconductanceofresponseofthesystemisdescribedbythefollowingmodifiede2/hratherthane2/2h,sincethedomainwallchiralstatebe-constituentequations,havesinthesamewayastheedgestateofaσ=e2/hQHHsystem.Thusagainweseethatfromdctransportmeasure-H=B−4πM+2P3αE,ments,onecannotobservethehalfHallconductance.D=E+4πP−2P3αB,(40)SuchadifferencebetweenintegerQHeffectandsurfacehalfQHeffectindicatesthatthesurfacehalfQHeffectiswithα=e2/~cthefinestructureconstant,andP≡3actuallyanewtopologicalphenomenonwhich,intermsofm/2|m|=±1/2thequantumofHallconductance.Ade-itsobservableconsequences,isqualitativelydifferentfromtailedexplanationofthecoefficientP3andtheeffectivefieldtheusualintegerQHeffect.Alternatively,theproperde-theorydescriptionoftheTMEeffectisdiscussedinSec.IV.tectionofthisnewtopologicalphenomenonactuallyprobesInthissectionwefocusonthephysicalconsequencesoftheauniqueelectromagneticresponsepropertyofthebulk,theTMEeffect.Wesimplynotethefactthatmoregenerally,forTME(Essinetal.,2009;Qietal.,2008).AmagnetoelectrictopologicalinsulatorsP3cantakethevaluen+1/2witharbi-effectisdefinedasamagnetizationinducedbyanelectrictraryintegern,sincethenumberofDiracconesonthesurfacefield,oralternatively,achargepolarizationinducedbyamag-canbeanyoddinteger.neticfield.TounderstandtherelationbetweensurfacehalfQHeffectandmagnetoelectriceffect,considertheconfigu-rationshowninFig.22(b),wherethesidesurfaceofa3D3.Imagemagneticmonopoleeffecttopologicalinsulatoriscoveredbymagneticimpuritieswithferromagneticorder,sothatthesurfaceisgappedandexhibitsOneofthemostdirectconsequencesoftheTMEeffectisahalfquantizedHallconductance.WhenanelectricfieldEtheimagemagneticmonopoleeffect(Qietal.,2009).Con-isappliedparalleltothesurface,aHallcurrentjisinducedsiderbringinganelectricchargetotheproximityofanordi-[Eq.(39)],whichcirculatesalongthesurface.Thissurfacenary3Dinsulator.Theelectricchargewillpolarizethedielec-currentperpendiculartoEwilltheninduceamagneticfieldtric,whichcanbedescribedbytheappearanceofanimageparalleltoE,sothatthesystemexhibitsamagnetoelectricelectricchargeinsidetheinsulator.Ifthesamethingisdoneresponse.TheHallresponseequationiswrittenaswithatopologicalinsulator,inadditiontotheimageelectric2chargeanimagemagneticmonopolewillalsoappearinsidemej=ˆn×E,(39)theinsulator.|m|2hThisimagemagneticmonopoleeffectcanbestudiedwithnˆaunitvectornormaltothesurface,andthesignofstraightforwardlybysolvingMaxwell'sequationswiththethemassm/|m|isdeterminedbythedirectionofthesurfacemodifiedconstituentequations(40),inthesamewayasthemagnetization.SuchaHallresponseisequivalenttoamag-imagechargeprobleminanordinaryinsulator.Considerthenetizationproportionaltotheelectricfield:geometryshowninFig.23(a).Thelowerhalf-spacez<02isoccupiedbyatopologicalinsulatorwithdielectricconstantmeMt=−E.ǫ2andmagneticpermeabilityµ2,whiletheupperhalf-space|m|2hcz>0isoccupiedbyaconventionalinsulatorwithdielec-Thismagnetizationisatopologicalresponsetotheelectrictricconstantǫ1andmagneticpermeabilityµ1.Anelectricfield,andisindependentofthedetailsofthesystem.Simi-pointchargeqislocatedat(0,0,d)withd>0.Weassumelarly,atopologicalcontributiontothechargepolarizationcanthatthesurfacestatesaregappedbysomelocalTRsymmetrybeinducedbyamagneticfield.Thecompleteelectromagneticbreakingfieldm,sothatthesurfacehalfQHeffectandTME 28exist.Theboundaryofthetopologicalinsulatoractsasado-zy,!mainwallwhereP3jumpsfrom1/2to0.Inthissemi-infinite11qxgeometry,animagepointmagneticmonopolewithfluxg2is(,qg11)(0,0,)dlocatedatthemirrorposition(0,0,−d),togetherwithanim-ageelectricpointchargeq2.Physically,themagneticfieldofxsuchanimagemonopoleconfigurationisinducedbycircu-latingHallcurrentsonthesurface,whichareinducedbytheelectricfieldoftheexternalcharge.AsimilareffecthasbeenstudiedintheintegerQHeffect(HaldaneandChen,1983).(,qg22)(0,0,"d)TI,!Conversely,theelectromagneticfieldstrengthinsidethetopo-22logicalinsulatorisdescribedbyanimagemagneticmonopoleg1andelectricchargeq1intheupperhalf-space,atthesameFIG.23(a)Imageelectricchargeandimagemagneticmonopolepointastheexternalcharge.Theimagemagneticmonopoleduetoanexternalelectricpointcharge.Thelowerhalf-spaceisfluxandimageelectriccharge(q1,g1)and(q2,g2)aregivenoccupiedbyatopologicalinsulator(TI)withdielectricconstantǫ2andmagneticpermeabilityµ2.Theupperhalf-spaceisoccupiedbybyatopologicallytrivialinsulator(e.g.vacuum)withdielectriccon-1(ǫ1−ǫ2)(1/µ1+1/µ2)−4α2P2stantǫ1andmagneticpermeabilityµ1.Anelectricpointchargeq3q1=q2=ǫ(ǫ+ǫ)(1/µ+1/µ)+4α2P2q,islocatedat(0,0,d).Seenfromthelowerhalf-space,theimage112123electricchargeq1andmagneticmonopoleg1areat(0,0,d).Seen4αP3fromtheupperhalf-space,theimageelectricchargeq2andmag-g1=−g2=−22q.(41)(ǫ1+ǫ2)(1/µ1+1/µ2)+4αP3neticmonopoleg2areat(0,0,−d).Thered(blue)solidlinesrepre-senttheelectric(magnetic)fieldlines.Theinsetisatop-downviewInterestingly,bymakinguseofelectric-magneticdualitytheseshowingthein-planecomponentoftheelectricfieldonthesurfaceexpressionscanbesimplifiedtomorecompactforms(Karch,(redarrows)andthecirculatingsurfacecurrent(blackcircles).(b)Il-2009).lustrationofthefractionalstatisticsinducedbytheimagemonopoleMoreover,interestingphenomenaappearwhenweconsidereffect.Eachelectronformsa“dyon”withitsimagemonopole.Whenthedynamicsoftheexternalcharge.Forexample,consideratwoelectronsareexchanged,aAharonov-Bohmphasefactorisob-tained,whichisdeter-minedbyhalfoftheimagemonopoleflux,2Delectrongasatadistancedabovethesurfaceofthe3Dindependentlyoftheexchangepath,leadingtothephenomenonoftopologicalinsulator.Ifthemotionoftheelectronisslowstatisticaltransmutation.FromQietal.,2009.enough(withrespecttothetimescale~/mcorrespondingtotheTRsymmetrybreakinggapm),theimagemonopolewillfollowtheelectronadiabatically,suchthattheelectronformsmayoccurforlightreflectedbyaTRsymmetrybreakinganelectron-monopolecomposite,i.e.adyon(Witten,1979).surface,whichisknownasthemagneto-opticalKerref-Whentwoelectronswindaroundeachother,eachelectronfect(LandauandLifshitz,1984).Sincethebulkofthetopo-perceivesthemagneticfluxoftheimagemonopoleattachedlogicalinsulatorisTRinvariant,noFaradayrotationwilloc-totheotherelectron,whichleadstostatisticaltransmutation.curinthebulk.However,ifTRsymmetryisbrokenontheThestatisticalangleisdeterminedbytheelectronchargeandsurface,theTMEeffectoccursandauniquekindofKerrandimagemonopolefluxasFaradayrotationisinducedonthesurface.Physically,the2planeofpolarizationofthetransmittedandreflectedlightisg1q2αP3θ=2~c=(ǫ+ǫ)(1/µ+1/µ)+4α2P2.(42)rotatedbecausetheelectricfieldE0(r,t)oflinearlypolarized12123lightgeneratesamagneticfieldB(r,t)inthesamedirection,TheimagemonopolecanbedetecteddirectlybylocalduetotheTMEeffect.Similarlyasfortheimagemonopoleprobessensitivetosmallmagneticfields,suchasscan-effect,theFaradayandKerrrotationanglescanbecalculatedningsuperconductingquantuminterferencedevices(scan-bysolvingMaxwell'sequationswiththemodifiedconstituentningSQUID)andscanningmagneticforcemicroscopyequations(40).Inthesimplestcaseofasinglesurfacebe-(scanningMFM)(Qietal.,2009).Thecurrentduetotweenatrivialinsulatorandasemi-infinitetopologicalinsu-theimagemonopolecanalsobedetectedinprinci-lator[Fig.24(a)],therotationangleforlightincidentfromple(ZangandNagaosa,2010).thetrivialinsulatorisgivenby(Karch,2009;Maciejkoetal.,2010;Qietal.,2008;TseandMacDonald,2010)p4.TopologicalKerrandFaradayrotation4αP3ǫ1/µ1tanθK=22,(43)ǫ2/µ2−ǫ1/µ1+4αP3AnotherwaytodetecttheTMEeffectisthroughthe2αP3transmissionandreflectionofpolarizedlight.Whenlin-tanθF=pp,(44)ǫ1/µ1+ǫ2/µ2earlypolarizedlightpropagatesthroughamediumwhichbreaksTRsymmetry,theplaneofpolarizationofthetrans-whereǫ1,µ1arethedielectricconstantandmagneticperme-mittedlightmayberotated,whichisknownastheFara-abilityofthetrivialinsulator,andǫ2,µ2arethoseofthetopo-dayeffect(LandauandLifshitz,1984).Asimilarrotationlogicalinsulator. 29sothatthemonopolecarriesanelectriccharge(Witten,1979)q=eθgwithφ=hc/ethefluxquanta.Suchacom-2πφ00positeparticlecarriesbothmagneticfluxandelectriccharge,qandiscalledadyon.Inprinciple,thetopologicalinsula-topotorprovidesaphysicalsystemwhichcandetectamagneticFMmonopolethroughthiseffect(RosenbergandFranz,2010).TIForatopologicalinsulator,wehaveθ=πwhichcorre-spondstoahalfchargeq=e/2foramonopolewithunitflux.Suchahalfchargecorrespondstoazeroenergyboundstateinducedbythemonopole.Thechargeofthemonopoleise/2whentheboundstateisoccupied,and−e/2whenitFIG.24(a)IllustrationoftheFaradayrotationononesurfaceofisunoccupied.Suchahalfchargeandzeromodeissimilartopologicalinsulator.(b)AmorecomplicatedgeometrywithKerrtochargefractionalizationin1Dsystems(Suetal.,1979).IfandFaradayrotationontwosurfacesoftopologicalinsulator.Inthisgeometry,theeffectofnon-universalpropertiesofthematerialcanbethemonopolepassesthroughaholeinthetopologicalinsula-eliminatedandthequantizedmagnetoelectriccoefficientαP3canbetor,thechargewillfollowit,whichcorrespondstoachargedirectlymeasured.FromQietal.,2008andMaciejkoetal.,2010.pumpingeffect(Rosenbergetal.,2010).AlleffectsdiscusseduptothispointareconsequencesoftheTMEinatopologicalinsulatorwithsurfaceTRsymmetryAlthoughthesurfaceFaradayandKerrrotationsarein-breaking.Noeffectsofelectroncorrelationhavebeentakenducedbythetopologicalpropertyofthebulk,andaredeter-intoaccount.Whentheelectron-electroninteractioniscon-minedbythemagnetoelectricresponsewithquantizedcoeffi-sidered,interestingneweffectscanoccur.Forexample,ifcientαP3,therotationangleisnotuniversalanddependsonatopologicalinsulatorisrealizedbytransitionmetalcom-thematerialparametersǫandµ.TheTMEresponsealwayspoundswithstrongelectroncorrelationeffects,antiferromag-coexistswiththeordinaryelectromagneticresponse,whichnetic(AFM)long-rangeordermaydevelopinthismaterial.makesitdifficulttoobservethetopologicalquantizationphe-SincetheAFMorderbreaksTRsymmetryandinversionsym-nomenon.However,recentlynewproposalshavebeenmademetry,themagnetoelectriccoefficientP3definedinEq.(40)toavoidthedependenceonnon-universalmaterialparame-deviatesfromitsquantizedvaluen+1/2.Denotingbyn(r,t)ters(Maciejkoetal.,2010;TseandMacDonald,2010).ThetheAFMN´eelvector,wehaveP3(n)=P3(n=0)+δP3(n)keyideaistoconsideraslaboftopologicalinsulatoroffi-whereP3(n=0)isthequantizedvalueofP3intheab-nitethicknesswithtwosurfaces,withvacuumononesidesenceofAFMorder.Thischangeinthemagnetoelectricco-andasubstrateontheother[Fig.24(b)].Thecombinationefficienthasinterestingconsequenceswhenspin-waveexcita-ofKerrandFaradayanglesmeasuredatreflectivityminimationsareconsidered.FluctuationsoftheN´eelvectorδn(r,t)providesenoughinformationtodeterminethequantizedcoef-induceingeneralfluctuationsofδP3,leadingtoacouplingficientαP3(Maciejkoetal.,2010),betweenspin-wavesandtheelectromagneticfield(Lietal.,cotθF+cotθK2010).Inhigh-energyphysics,suchaparticlecoupledto2=2αP3,(45)1+cotθFtheE·Btermiscalledan“axion”(PecceiandQuinn,1977;Wilczek,2009).Physically,inabackgroundmagneticfieldprovidedthatbothtopandbottomsurfaceshavethesamesur-2suchan“axionic”spin-waveiscoupledtotheelectricfieldfaceHallconductanceσH=P3e/h.InEq.(45),thequan-withacouplingconstanttunablebythemagneticfield.Con-tizedcoefficientαP3isexpressedsolelyintermsofthemea-sequently,apolaritoncanbeformedbythehybridizationofsurableKerrandFaradayangles.Thisenablesadirectexper-thespin-waveandphoton,similartothepolaritonformedbyimentalmeasurementofP3withoutaseparatemeasurementopticalphonons(MillsandBurstein,1974).Thepolaritongapofthenon-universalopticalconstantsǫ,µofthetopologicaliscontrolledbythemagneticfield,whichmayrealizeatun-insulatorfilmandthesubstrate.Ifthetwosurfaceshavedif-ableopticalmodulator.ferentsurfaceHallconductances,itisstillpossibletodeter-Anotherinterestingeffectemergesfromelectroncorrela-minethemseparatelythroughameasurementoftheKerrandtionswhenathinfilmoftopologicalinsulatorisconsidered.Faradayanglesatreflectivitymaxima(Maciejkoetal.,2010).Whenthefilmisthickenoughsothatthereisnodirecttun-nelingbetweenthesurfacestatesonthetopandbottomsur-5.Relatedeffectsfaces,butnottoothicksothatthelong-rangeCoulombinter-actionbetweenthetwosurfacesarestillimportant,ainter-TheTMEhasotherinterestingconsequences.TheTMEef-surfaceparticle-holeexcitation,i.e.anexciton,canbein-fectcorrespondstoatermθE·Bintheaction(seeSec.IVforduced(Seradjehetal.,2009).Denotingthefermionannihi-moredetails),whichmediatesthetransmutationbetweenelec-lationoperatoronthetwosurfacesbyψ1,ψ2,theexciton†tricfieldandmagneticfield(Qietal.,2008).Inthepresencecreationoperatorisψ1ψ2.Inparticular,whenthetwosur-ofamagneticmonopole,suchanelectric-magnetictransmu-faceshaveoppositeFermienergywithrespecttotheDiractationinducesanelectricfieldaroundthemagneticmonopole,point,thereisnestingbetweenthetwoFermisurfaces,which 30leadstoaninstabilitytowardsexcitoncondensation.Inthefrombulksamples(Hongetal.,2010;Shahiletal.,2010;excitoncondensatephase,theexcitoncreationoperatorac-Teweldebrhanetal.,2010).Thestoichiometriccompounds†quiresanonzeroexpectationvaluehψ1ψ2i6=0,whichcor-Bi2Se3,Bi2Te3,Sb2Te3arenotextremelydifficulttogrow,respondstoaneffectiveinter-surfacetunneling.Interest-whichshouldallowmoreexperimentalgroupstohaveaccessingly,onecanconsideravortexinthisexcitoncondensate.tohigh-qualitytopologicalinsulatorsamples(Butchetal.,Suchavortexcorrespondstoacomplexspatially-dependent2010;Zhangetal.,2009).duetointrinsicdopingfromva-inter-surfacetunnelingamplitude,andisequivalenttoamag-cancyandanti-sitedefects,Bi2Se3andBi2Te3(Chenetal.,neticmonopole.AccordingtotheWitteneffectmentioned2009;Hsiehetal.,2009;Xiaetal.,2009)areshowntocon-above(RosenbergandFranz,2010),suchavortexoftheex-tainn-typecarrierswhileSb2Te3(Hsiehetal.,2009)isp-citoncondensatecarriescharge±e/2,whichprovidesawaytype.Consequently,controllableextrinsicdopingisre-totesttheWitteneffectintheabsenceofarealmagneticquiredtotunetheFermienergytotheDiracpointofmonopole.thesurfacestates.Forexample,Bi2Te3canbedopedBesidestheeffectsdiscussedabove,therearemanyotherwithSn(Chenetal.,2009)andBi2Se3canbedopedwithphysicaleffectsrelatedtotheTMEeffect,orsurfacehalfQHSb(Analytisetal.,2010)orCa(Horetal.,2010;Wangetal.,effect.Whenamagneticlayerisdepositedontopofthetopo-2010).Furthermore,itisfoundthatdopingBi2Se3withlogicalinsulatorsurface,thesurfacestatescanbegappedandCucaninducesuperconductivity(Horetal.,2010),whileahalfQHeffectisinduced.Inotherwords,themagneticFeandMndopantsmayyieldferromagnetism(Chaetal.,momentofthemagneticlayerdeterminestheHallresponse2010;Chenetal.,2010;Horetal.,2010;Wrayetal.,2010;ofthesurfacestates,whichcanbeconsideredasacouplingXiaetal.,2008).betweenthemagneticmomentandthesurfaceelectriccur-rent(Qietal.,2008).SuchacouplingleadstotheinverseofthehalfQHeffect,whichmeansthatachargecurrenton2.Angle-resolvedphotoemissionspectroscopythesurfacecanflipthemagneticmomentofthemagneticlayer(GarateandFranz,2010).Similartosuchacouplingbe-ARPESexperimentsareuniquelypositionedtodetectthetweenchargecurrentandmagneticmoment,achargedensitytopologicalsurfacestates.Thefirstexperimentsontopo-iscoupledtomagnetictexturessuchasdomainwallsandvor-logicalinsulatorswereARPESexperimentscarriedoutontices(NomuraandNagaosa,2010).ThiseffectcanbeusedtheBi1−xSbxalloy(Hsiehetal.,2008).Theobservationtodrivemagnetictexturesbyelectricfields.Theseeffectsonoffivebranchesofsurfacestates,togetherwiththere-atopologicalinsulatorsurfacecoupledwithmagneticlayersspectivespinpolarizationsdeterminedlaterbyspin-resolvedarerelevanttopotentialapplicationsoftopologicalinsulatorsARPES(Hsiehetal.,2009),confirmsthenontrivialtopologi-indesigningnewspintronicsdevices.calnatureofthesurfacestatesofBi1−xSbx.ARPESworkonBi2Se3(Xiaetal.,2009)andBi2Te3(Chenetal.,2009;Hsiehetal.,2009)soonfollowed.E.ExperimentalresultsUnlikethemultiplebranchesofsurfacestatesobservedforBi1−xSbx,theseexperimentsreportaremarkablysimple1.MaterialgrowthsurfacestatespectrumwithasingleDiracconelocatedattheΓpointandalargebulkbandgap,inaccordancewiththeTherehavebeenmanyinterestingtheoreticalproposalsfortheoreticalpredictions.ForBi2Se3,asingleDiracconewithnoveleffectsintopologicalinsulators,butperhapsthemostlineardispersionisclearlyshownattheΓ¯pointwithintheexcitingaspectofthefieldistherapidincreaseinexperimen-bandgapinFig.25(a)and(b).Figure25(d)showstheycom-taleffortsfocussedontopologicalinsulators.High-qualityponentofthespinpolarizationalongthekx(Γ¯−M¯)directionmaterialsarebeingproducedinseveralgroupsaroundthemeasuredbyspin-resolvedARPES(Hsiehetal.,2009).Theworld,andofalldifferenttypes.BulkmaterialswerefirstoppositespinpolarizationintheydirectionforoppositekgrownforexperimentsontopologicalinsulatorsintheCavaindicatesthehelicalnatureofthespinpolarizationforsurfacegroupatPrincetonUniversityincludingtheBi1−xSbxal-states.Asdiscussedabove,Bi2Se3hasafinitedensityofloy(Hsiehetal.,2008)andBi2Se3,Bi2Te3,Sb2Te3crys-n-typecarriersduetointrinsicdoping.Therefore,theabovetals(Hsiehetal.,2009;Xiaetal.,2009).Crystallinesam-ARPESdata[Fig.25(a),(b)]showsthattheFermienergyisplesofBi2Te3havealsobeengrownatStanfordUniver-abovetheconductionbandbottomandthesampleis,infact,sityintheFishergroup(Chenetal.,2009).Inadditionametalratherthananinsulatorinthebulk.Toobtainatruetobulksamples,Bi2Se3nanoribbons(Hongetal.,2010;topologicalinsulatingstatewiththeFermienergytunedintoKongetal.,2010;Pengetal.,2010)havebeenfabricatedthebulkgap,carefulcontrolofexternaldopingisrequired.intheCuigroupatStanfordUniversity,andthinfilmsofSuchcontrolwasfirstreportedbyChenetal.[Fig.26]foraBi2Se3andBi2Te3havebeengrownbyMBEbytheXuesampleofBi2Te3with0.67%Sndoping(Chenetal.,2009).groupatTsinghuaUniversity(Lietal.,2009;Zhangetal.,SomerecentworkonSb2Te3(Hsiehetal.,2009)supportsthe2009),aswellasothergroups(Lietal.,2010;Zhangetal.,theoreticalpredictionthatthismaterialisalsoatopological2009).Thinfilmscanalsobeobtainedbyexfoliationinsulator(Zhangetal.,2009).Thisfamilyofmaterialsis 31FIG.26ARPESmeasurementof(a)shapeoftheFermisurfaceand(b)banddispersionalongtheK−Γ−Kdirection,forBi2Te3nom-inallydopedwith0.67%Sn.FromChenetal.,2009.FIG.25ARPESdataforthedispersionofthesurfacestatesofBi2Se3,alongdirections(a)Γ¯−M¯and(b)Γ¯−K¯inthesurfaceBrilliounzone.Spin-resolvedARPESdataisshownalongΓ¯−M¯forafixedenergyin(d),fromwhichthespinpolarizationinmomentumspace(c)canbeextracted.FromXiaetal.,2009andHsiehetal.,FIG.27ARPESdataforBi2Se3thinfilmsofthickness(a)1QL(b)2009.2QL(c)3QL(d)5QL(e)6QL,measuredatroomtemperature(QLstandsforquintuplelayer).FromZhangetal.,2009.movingtotheforefrontofresearchontopologicalinsulatorsduetothelargebulkgapandthesimplicityofthesurface3.Scanningtunnelingmicroscopystatespectrum.InadditiontotheARPEScharacterizationof3Dtopolog-Althoughthesimplemodel(101)capturesmostoftheicalinsulators,scanningtunnelingmicroscopy(STM)andsurfacestatephysicsofthesesystems,experimentsreportscanningtunnelingspectroscopy(STS)provideanotherkindahexagonalsurfacestateFermisurface[Fig.26],whileofsurface-sensitivetechniquetoprobethetopologicalsur-Eq.(101)onlydescribesacircularFermisurfacesufficientlyfacestates.AsetofmaterialshavebeeninvestigatedinclosetotheDiracpoint.However,suchahexagonalwarpingSTM/STSexperiments:Bi1−xSbx(Roushanetal.,2009),effectcanbeeasilytakenintoaccountbyincludinganaddi-Bi2Te3(Alpichshevetal.,2010;Zhangetal.,2009),andtionalterminthesurfaceHamiltonianwhichiscubicink(Fu,Sb(Gomesetal.,2009).(AlthoughSbistopologicallynon-2009).ThesurfaceHamiltonianforBi2Te3canbewrittentrivial,itisasemi-metalinsteadofaninsulator.)Thecompar-isonbetweenSTM/STSandARPESwasfirstperformedforyxλ33zBiTe(Alpichshevetal.,2010),whereitwasfoundthattheH(k)=E0(k)+vk(kxσ−kyσ)+(k++k−)σ,(46)232integrateddensityofstatesobtainedfromARPES[Fig.28(a)]agreeswellwiththedifferentialconductancedI/dVobtainedwhereE(k)=k2/(2m∗)breakstheparticle-holesymmetry,0fromSTSmeasurements[Fig.28(b)].Fromsuchacompar-theDiracvelocityv=v(1+αk2)acquiresaquadraticde-kison,differentcharacteristicenergies(EF,EA,EB,ECandpendenceonk,andλparameterizestheamountofhexagonalEDinFig.28)canbeeasilyandunambiguouslyidentified.warping(Fu,2009).BesidesthelinearDiracdispersionwhichhasalreadybeenInadditiontoitsusefulnessforstudyingbulkcrystallinewellestablishedbyARPESexperiments,STM/STScanpro-samples,ARPEShasalsobeenusedtocharacterizethethinvidefurtherinformationaboutthetopologicalnatureofthefilmsofBi2Se3andBi2Te3(Lietal.,2009;Sakamotoetal.,surfacestates,suchastheinterferencepatternsofimpuri-2010;Zhangetal.,2009).Thethinfilmsweregrowntoiniti-tiesoredges(Alpichshevetal.,2010;Gomesetal.,2009;ateastudyofthecrossover(Liuetal.,2010)froma3Dtopo-Roushanetal.,2009;Zhangetal.,2009).Whentherearelogicalinsulatortoa2DQSHstate(Sec.III.C).InFig.27,impuritiesonthesurfaceofatopologicalinsulator,thesur-ARPESspectraareshownforseveralthicknessesofaBi2Se3facestateswillbescatteredandformaninterferencepat-thinfilm,whichshowtheevolutionofthesurfacestates.ternaroundtheimpurities.Fouriertransformingtheinter- 32FIG.29(a)MeasuredinterferencepatterninmomentumspaceforimpuritiesonthesurfaceofBixSb1−x.(b)PatterncalculatedfromARPESdataonBixSb1−x,whichagreeswellwiththeinterferencepatternin(a).(c)Similarinterferencepatternand(d)possiblescat-teringwavevectorsforBi2Te3.FromRoushanetal.,2009andFIG.28Goodagreementisfoundbetween(a)theintegratedden-Zhangetal.,2009.sityofstatesfromARPESand(b)atypicalscanningtunnelingspec-troscopyspectrum.EFistheFermilevel,EAthebottomofthebulkconductionband,EBthepointwherethesurfacestatesbecomewarped,ECthetopofthebulkvalenceband,andEDtheDiracistheobservationofsurfacestateLLsinamagneticpoint.FromAlpichshevetal.,2010.field(Chengetal.,2010;Hanagurietal.,2010).AsshowninFig.30(a)and(b),discreteLLsappearasaseriesofpeaksinthedifferentialconductancespectrum(dI/dV),whichsup-ferencepatternintomomentumspace,onecanquantitativelyportsthe2Dnatureofthesurfacestates.Furtheranalysisonextractthescatteringintensityforafixedenergyandscatter-thedependenceoftheLLsonthemagneticfield√Bshowsthatingwavevector.WithsuchinformationonecandeterminetheenergyoftheLLsisproportionaltonBwherenisthewhattypesofscatteringeventsaresuppressed.Figure29(a)Landaulevelindex,insteadoftheusuallinear-in-Bdepen-and(c)showstheinterferencepatterninmomentumspacefordence.ThisunusualdependenceprovidesadditionalevidenceBixSb1−x(Roushanetal.,2009)andBi2Te3(Zhangetal.,fortheexistenceofsurfacestatesconsistingofmasslessDirac2009),respectively.Inordertoanalyzetheinterferencefermions.Furthermore,thenarrowpeaksinthespectrumalsopattern(Leeetal.,2009),wetakeBi2Te3asanexampleindicatethegoodqualityofthesamplesurface.[Fig.29(c),(d)].ThesurfaceFermisurfaceofBi2Te3isshowninFig.29(d),forwhichthepossiblescatteringeventsaredom-inatedbythewavevectorsq1alongtheK¯direction,q2along4.TransporttheM¯directionandq3betweentheK¯andM¯directions.However,fromFig.29(c)weseethatthereisapeakalongInadditiontotheabovesurface-sensitivetechniques,theΓ¯−M¯direction,whilescatteringalongtheΓ¯−K¯di-alargeefforthasbeendevotedtotransportmeasure-rectionissuppressed.Thisobservationcoincideswiththementsincludingdctransport(Analytisetal.,2010;theoreticalpredictionthatbackscatteringbetweenkand−kButchetal.,2010;Chenetal.,2010;Etoetal.,2010;isforbiddenduetoTRsymmetry,whichsupportsthetopo-Steinbergetal.,2010;Tangetal.,2010)andmeasure-logicalnatureofthesurfacestates.Otherrelatedtheoreti-mentsinthemicrowave(Analytisetal.,2010)andin-calanalysisarealsoconsistent(BiswasandBalatsky,2010;fraredregimes(Butchetal.,2010;LaForgeetal.,2010;GuoandFranz,2010;Zhouetal.,2009).AsimilaranalysisSushkovetal.,2010),whicharenecessarystepstowardscanbeappliedtothesurfaceofBixSb1−x,andtheobtainedthedirectmeasurementoftopologicaleffectssuchasthepattern[Fig.29(b)]alsoagreeswellwiththeexperimentalTME,andforfuturedeviceapplications.However,transportdata[Fig.29(a)](Roushanetal.,2009).Morerecently,STMexperimentsontopologicalinsulatorsturnouttobemuchexperimentshavefurtherdemonstratedthatthetopologicalmoredifficultthansurface-sensitivemeasurementssuchsurfacestatescanpenetratebarrierswhilemaintainingtheirasARPESandSTM.Themaindifficultyarisesfromtheextendednature(Seoetal.,2010).existenceofafiniteresidualbulkcarrierdensity.MaterialsAnotherimportantresultofSTM/STSmeasurementssuchasBi1−xSbxorBi2Se3arepredictedtobetopological 33FIG.30TunnelingspectraforthesurfaceofBi2Se3inamagneticFIG.31Angulardependenceofthecyclotronresonancefieldforafield,showingaseriesofpeaksattributedtotheoccurrenceofsurface71GHzmicrowave.Theinsetshowsexamplesoftransmissiondata◦◦◦Landaulevels.FromChengetal.,2010andHanagurietal.,2010.offsetforclarity(toptobottom:90to0instepsof10).FromAyala-Valenzuelaetal.,2010.insulatorsiftheyareperfectlycrystalline.However,realmaterialsalwayshaveimpuritiesanddefectssuchasanti-sitesfects(Checkelskyetal.,2010;Chenetal.,2010).However,andvacancies.Therefore,as-grownmaterialsarenottrulybesidestheexperimentsmentionedabovewithpositiveev-insulatingbuthaveafinitebulkcarrierdensity.Asdiscussedidencefortheexistenceoftopologicalsurfacestates,someinSec.III.E.2,sucharesidualbulkcarrierdensityisalsoexperimentsshowthatthetransportdatacanbeentirelyex-observedinARPESforBi2Se3andBi2Te3(Chenetal.,plainedbybulkcarriers(Butchetal.,2010;Etoetal.,2010).2009;Hsiehetal.,2009).FromtheARPESresultsitseemsTheresolutionofthiscontroversyrequiresfurtherimprove-thattheresidualcarrierdensitycanbecompensatedforbymentsinexperimentsandsamplequality.chemicaldoping(Chenetal.,2009).Nevertheless,intrans-Toreachtheintrinsictopologicalinsulatorstatewithoutportexperimentsthecompensationofbulkcarriersappearstobulkcarriers,variouseffortshavebeenmadetoreducethebemuchmoredifficult.Evensampleswhichappearasbulkbulkcarrierdensity.Oneapproachconsistsincompensat-insulatorsinARPESexperimentsstillexhibitsomefinitebulkingthebulkcarriersbychemicaldoping,e.g.dopingBi2Se3carrierdensityintransportmeasurements(Analytisetal.,withSb(Analytisetal.,2010),Ca(Horetal.,2009),ordop-2010),whichsuggeststheexistenceofanoffsetbetweeningBi2Te3withSn(Chenetal.,2009).Althoughchem-bulkandsurfaceFermilevels.Anotherdifficultyintransporticaldopingisanefficientwaytoreducethebulkcarriermeasurementsisthatacleavedsurfacerapidlybecomesdensity,themobilitywillbeusuallyreducedduetoforeignheavilyn-dopedwhenexposedtoair.Thisleadstofurtherdopants.However,wenotethatthesubstitutionoftheisova-discrepanciesbetweenthesurfaceconditionobservedinlentBiwithSbcanreducethecarrierdensitybutstillkeeptransportandsurface-sensitivemeasurements.highmobilities(Analytisetal.,2010).Also,itisdifficultInspiteofthecomplexitydescribedabove,thesignaturetoachieveaccuratetuningofthecarrierdensitybychem-of2Dsurfacestatesintransportexperimentshasbeenre-icaldoping,becauseeachdifferentchemicaldopinglevelcentlyreported(Analytisetal.,2010;Ayala-Valenzuelaetal.,needstobereachedbygrowinganewsample.Thesec-2010).Forexample,Fig.31showsresultsobtainedondmethodconsistsinsuppressingthecontributionofbulkbymicrowavespectroscopy(Ayala-Valenzuelaetal.,2010)carrierstotransportbyreducingthesamplesizedowntoonBi2Se3,whereitisfoundthatthecyclotronres-thenanoscale,suchasquasi-1Dnanoribbons(Kongetal.,onancefrequencyonlyscaleswithperpendicularmag-2010;Pengetal.,2010;Steinbergetal.,2010),orquasi-2DneticfieldB⊥,suggestingthe2Dnatureofthereso-thinfilm(Checkelskyetal.,2010;Chenetal.,2010;Lietal.,nance.Similarly,thedependenceofShubnikov-deHaas2009).InFig.32,themagnetoresistanceofananorib-oscillationsontheangleofthemagneticfieldcanhelpbonexhibitsaprimaryhc/eoscillation,whichcorrespondstodistinguishthe2Dsurfacestatesfromthe3DbulktoAharonov-Bohmoscillationsofthesurfacestatearoundstates,bothforBi2Se3(Ayala-Valenzuelaetal.,2010)andthesurfaceofthenanoribbon(Pengetal.,2010).ThisBixSb1−x(TaskinandAndo,2009).Signaturesofthetopo-oscillationalsoindicatesthatthebulkcarrierdensityhaslogicalsurfacestateshavealsobeensearchedforinthebeenreducedgreatlysothatthecontributionofthesurfacetemperaturedependenceoftheresistance(Analytisetal.,statescanbeobserved.TheAharonov-Bohmoscillationhas2010;Checkelskyetal.,2010,2009),themagnetoresis-alsobeeninvestigatedtheoretically(Bardarsonetal.,2010;tance(Tangetal.,2010),andweakantilocalizationef-ZhangandVishwanath,2010).Animportantadvantageof 34(a)(b)FIG.33(a)Gatevoltagedependenceoftheresistanceinzeromag-neticfieldfordifferenttemperatures.(b)GatevoltagedependenceoftheHallresistanceRyxinamagneticfieldof5Tfordifferenttemperatures.FromCheckelskyetal.,2010.trasttothelayeredtetradymitecompounds.ThesematerialsFIG.32Magnetoresistanceforfieldsupto±9T.Leftinset:haverecentlybeenexperimentallyobservedtobetopologicalmagneticfieldsatwhichwell-developedresistanceminimaareob-insulators(Chenetal.,2010;Satoetal.,2010).served.Rightinset:fastFouriertransformoftheresistancederiva-AtypicalmaterialofthesecondgroupisdistortedbulktivedR/dB,wherepeakscorrespondtohc/eandhc/2eoscillationsarelabeled.FromPengetal.,2010.HgTe.Incontrasttoconventionalzincblendesemiconductors,HgTehasaninvertedbulkbandstructurewiththeΓ8bandbeinghigherinenergythantheΓ6band.However,HgTebyasampleofmesoscopicsizeisthepossibilityoftuningtheitselfisasemi-metalwiththeFermienergyatthetouchingcarrierdensitybyanexternalgatevoltage.Gatecontrolofpointbetweenthelight-holeandheavy-holeΓ8bands.Con-thecarrierdensityisratherimportantbecauseitcantunethesequently,inordertogetatopologicalinsulator,thecrys-bulkcarrierdensitycontinuouslywhilepreservingthequalitytalstructureofHgTeshouldbedistortedalongthe[111]di-ofthesample.Gatecontrolofthecarrierdensityhasbeenrectiontoopenagapbetweentheheavy-holeandlight-holeindeedobservedinnanoribbonsofBi2Se3(Steinbergetal.,bands(Daietal.,2008).Asimilarbandstructurealsoexists2010),mechanicallyexfoliatedthinfilms(Checkelskyetal.,internaryHeuslercompounds(Chadovetal.,2010;Linetal.,2010),orepitaxiallygrownthinfilms(Chenetal.,2010).In2010),andaroundfiftyofthemarefoundtoexhibitbandin-particular,tuningofthecarrierpolarityfromn-typetop-typeversion.Thesematerialsbecome3Dtopologicalinsulatorshasbeenreported(Checkelskyetal.,2010),wherethechangeupondistortion,ortheycanbegrowninquantumwellforminpolaritycorrespondstoasignchangeoftheHallresistancesimilartoHgTe/CdTetorealizethe2DortheQSHinsula-Ryxinamagneticfield[Fig.33(b)].tors.DuetothediversityofHeuslermaterials,multifunc-tionaltopologicalinsulatorscanberealizedwithadditionalpropertiesrangingfromsuperconductivitytomagnetismandF.Othertopologicalinsulatormaterialsheavy-fermionbehavior.Besidestheabovetwolargegroupsofmaterials,thereareThetopologicalmaterialsHgTe,Bi2Se3,Bi2Te3andalsosomeothertheoreticalproposalsofnewtopologicalinsu-Sb2Te3notonlyprovideuswithaprototypematerialfor2Dlatormaterialswithelectroncorrelationeffects.Anexampleand3Dtopologicalinsulators,butalsogiveusaruleofthumbisthecaseofIr-basedmaterials.TheQSHeffecthasbeenpro-tosearchfornewtopologicalinsulatormaterials.Thenon-posedinNa2IrO3(Shitadeetal.,2009),andtopologicalMotttrivialtopologicalpropertyoftopologicalinsulatorsoriginatesinsulatorphaseshavebeenproposedinIr-basedpyrochlorefromtheinvertedbandstructureinducedbySOC.Therefore,oxidesLn2Ir2O7withLn=Nd,Pr(GuoandFranz,2009;itismorelikelytofindtopologicalinsulatorsinmaterialsPesinandBalents,2010;Wanetal.,2010;YangandKim,whichconsistofcovalentcompoundswithnarrowbandgaps2010).Furthermore,atopologicalstructurehasalsobeenandheavyatomswithstrongSOC.FollowingsuchaguidingconsideredinKondoinsulators,withapossiblerealizationinprinciple,alargenumberoftopologicalinsulatormaterialsSmB6andCeNiSn(Dzeroetal.,2010).havebeenproposedrecently,whichcanberoughlyclassifiedintoseveraldifferentgroups.Thefirstgroupissimilartothetetradymitesemiconduc-IV.GENERALTHEORYOFTOPOLOGICALINSULATORStors,wheretheatomicp-orbitalsofBiorSbplayanessentialrole.Thallium-basedIII-V-VI2ternarychalcogenides,includ-TheTFT(Qietal.,2008)andtheTBT(FuandKane,2007;ingTlBiQ2andTlSbQ2withQ=Te,SeandS,belongtothisFuetal.,2007;KaneandMele,2005;MooreandBalents,class(Linetal.,2010;Yanetal.,2010).Thesematerialshave2007;Roy,2009)aretwodifferentgeneraltheoriesofthethesamerhombohedralcrystalstructure(spacegroupD5)astopologicalinsulators.TheTBTisvalidforthenon-3dthetetradymitesemiconductors,butaregenuinely3D,incon-interactingsystemwithoutdisorder.TheTBThasgivensim- 35pleandimportantcriteriatoevaluatewhichbandinsulatorscanbecalculatedexplicitlyfromasingleFeynmandiagramaretopologicallynon-trivial.TheTFTisgenerallyvalidfor[Fig.34(a)](Goltermanetal.,1993;NiemiandSemenoff,interactingsystemsincludingdisorder,andtheitidentifiesthe1983;Qietal.,2008;Volovik,2002),andoneobtainsC1asphysicalresponseassociatedwiththetopologicalorder.Re-giveninEq.(48),butwithGreplacedbythenoninteract-markably,theTFTreducesexactlytotheTBTinthenon-ingGreen'sfunctionG0.Carryingoutthefrequencyintegralinteractinglimit.Inthissection,wereviewbothgeneraltheo-explicitly,oneobtainstheTKNNinvariantexpressedasanries,andalsodiscusstheirconnections.integraloftheBerrycurvature(Thoulessetal.,1982),ZZ1C1=dkxdkyfxy(k)∈Z,(50)A.Topologicalfieldtheory2πwithWearegenerallyinterestedinthelong-wavelengthandlow-∂ay(k)∂ax(k)energypropertiesofacondensedmattersystem.Inthiscase,fxy(k)=−,∂kx∂kythedetailsofthemicroscopicHamiltonianarenotimportant,X∂andwewouldliketocaptureessentialphysicalpropertiesinai(k)=−ihαk||αki,i=x,y.termsofalow-energyeffectivefieldtheory.Forconventionalα∈occ∂kibroken-symmetrystates,thelow-energyeffectivefieldtheoryUnderTR,wehaveA0→A0,A→−A,fromwhichweisfullydeterminedbytheorderparameter,symmetryanddi-seethatthe(2+1)DCSfieldtheoryinEq.(47)breaksTRmensionality(Anderson,1997).Topologicalstatesofquan-symmetry.Allthelow-energyresponseoftheQHsystemcantummatteraresimilarlydescribedbyalow-energyeffectivebederivedfromthisTFT.Forinstance,fromtheeffectiveLa-fieldtheory.Inthiscase,theeffectivefieldtheorygenerallygrangianinEq.(47),takingafunctionalderivativewithre-involvetopologicaltermswhichdescribetheuniversaltopo-spectivetoAµ,weobtainthecurrentlogicalpropertiesofthestate.Thecoefficientofthetopolog-icaltermcanbegenerallyidentifiedasthetopologicalorderC1µντjµ=ǫ∂νAτ.(51)parameterofthesystem.AsuccessfulexampleistheTFT2πoftheQHeffect(Zhang,1992),whichcapturestheuniver-ThespatialcomponentofthiscurrentisgivenbysaltopologicalpropertiessuchasthequantizationoftheHallC1ijconductance,thefractionalchargeandstatisticsofthequasi-ji=ǫEj,(52)2πparticlesandthegroundstatedegeneracyonatopologicallynontrivialspatialmanifold.Inthissection,weshalldescribewhilethetemporalcomponentisgivenbytheTFToftheTRinvarianttopologicalinsulators.C1ijC1j0=ǫ∂iAj=B.(53)2π2π1.Chern-Simonsinsulatorin2+1dimensionsThisisexactlytheQHresponsewithHallconductanceσH=C1/(2π),implyingthatanelectricfieldinducesatransverseWestartfromthepreviouslymentionedQHsystemin(2+current,andamagneticfieldinduceschargeaccumulation.1)D,theTFTforwhichisgivenas(Zhang,1992)TheMaxwelltermcontainsmorederivativesthantheCStermZZandisthereforelessrelevantatlowenergiesintherenormal-C12µντizationgroupsense.Therefore,allthetopologicalresponseSeff=dxdtAµǫ∂νAτ,(47)4πoftheQHstateisexactlycontainedinthelow-energyTFTofwherethecoefficientC1isgenerallygivenby(Wangetal.,Eq.(47).2010)Zπd3kC=Tr[ǫµνρG∂G−1G∂G−1G∂G−1](48),2.Chern-Simonsinsulatorin4+1dimensions13(2π)3µνρandG(k)≡G(k,ω)istheimaginary-timesingle-particleTheTFToftheQHeffectdoesnotonlycapturetheuniver-Green'sfunctionofafullyinteractinginsulator,andµ,ν,ρ=sallow-energyphysics,italsopointsoutawaytogeneralize0,1,2≡t,x,y.Forageneralinteractingsystem,assumingtheTRsymmetrybreakingQHstatetoTRinvarianttopolog-thatGisnonsingular,wehaveamapfromthethreedimen-icalstates.TheCSfieldtheorycanbegeneralizedtoalloddsionalmomentumspacetothespaceofnonsingularGreen'sdimensionalspacetimes(Nakahara,1990).Thisobservationfunctions,belongingtothegroupGL(n,C),whosethirdho-leadZhangandHutodiscoverageneralizationoftheQHin-motopygroupislabeledbyaninteger(Wangetal.,2010):sulatorstate(ZhangandHu,2001)whichisTRinvariant,anddefinedin(4+1)D.ItisthefundamentalTRinvariantin-π3(GL(n,C))∼=Z.(49)sulatorstatefromwhichallthelower-dimensionalcasesareThewindingnumberforthishomotopyclassisexactlymea-derived,andisdescribedbytheTFT(Bernevigetal.,2002)ZsuredbyC1definedinEq.(48).Heren≥3isthenum-C24µνρστberofbands.Inthenoninteractinglimit,C1inEq.(47)Seff=24π2dxdtǫAµ∂νAρ∂σAτ.(54) 36wherex,y,z,warespatialcoordinatesandtistime.TheonlynonvanishingcomponentsofthefieldstrengthareFxy=BzandFzt=−Ez.AccordingtoEq.(58),thisfieldconfigura-tioninducesthecurrentC2jw=2BzEz.4πIfweintegratetheequationaboveoverthex,ydimensions,withperiodicboundaryconditionsandassumingthatEzdoesFIG.34FermionloopdiagramsleadingtotheChern-Simonsterm.notdependonx,y,weobtain(a)The(2+1)DChern-Simonstermiscalculatedfromaloopdia-ZZgramwithtwoexternalphotonlines.(b)The(4+1)DChern-SimonsC2C2Nxytermiscalculatedfromaloopdiagramwiththreeexternalphotondxdyjw=2dxdyBzEz≡Ez,(60)4π2πlines.RwhereNxy=dxdyBz/2πisthenumberoffluxquantathroughthexyplane,whichisalwaysquantizedtobeanin-UnderTR,wehaveA0→A0,A→−A,andthistermisteger.Thisisexactlythe4DgeneralizationoftheQHeffectexplicitlyTRinvariant.Generally,thecoefficientC2isex-mentionedearlier(ZhangandHu,2001).Therefore,fromthispressedintermsoftheGreen'sfunctionofaninteractingsys-examplewecanunderstandthephysicalresponseassociatedtemas(Wangetal.,2010)withanonvanishingsecondChernnumber.Ina(4+1)Din-Zπ2d5ksulatorwithsecondChernnumberC2,aquantizedHallcon-C≡Tr[ǫµνρστG∂G−1G∂G−1G∂G−1215(2π)5µνρductanceC2Nxy/2πinthezwplaneisinducedbyamagnetic−1−1fieldwithflux2πNxyintheperpendicular(xy)plane.×G∂σGG∂τG],(55)WehavediscussedtheCSinsulatorsin(2+1)Dandwhichlabelsthehomotopygroup(Wangetal.,2010)(4+1)D.Actually,thesediscussionscanbestraightforwardlygeneralizedtohigherdimensions.Indoingso,itisworthπ5(GL(n,C))∼=Z,(56)notingthatthereisaeven-oddalternationofthehomotopysimilarlytothecaseofthe(2+1)DCSterm.Foranonin-groupsofGL(n,C):wehaveπ2k+1(GL(n,C))∼=Z,whileteractingsystem,C2canbecalculatedfromasingleFeynmanπ2k(GL(n,C))=0.Thisisthemathematicalmechanismdiagram[Fig.34(b)]andoneobtainsC2asgiveninEq.(55),underlyingthefactthatCSinsulatorsofintegerclassexistinwithGreplacedbythenoninteractingGreen'sfunctionG0.evenspatialdimensions,butdonotexistinoddspatialdi-ExplicitintegrationoverthefrequencygivesthesecondChernmensions.Reducedtononinteractinginsulators,thisbecomesnumber(Qietal.,2008),thealternationofChernnumbers.AsanumbercharacteristicZofcomplexfiberbundles,Chernnumbersexistonlyineven14ijkℓspatialdimensions.ThisisanexampleoftherelationshipC2=2dkǫtr[fijfkℓ],(57)32πbetweenhomotopytheoryandhomologytheory.Weshallwithseeanotherexampleofthisrelationshipinthefollowingsec-αβαβαβαβtion:boththeWess-Zumino-Witten(WZW)termsandtheCSfij=∂iaj−∂jai+i[ai,aj],termsarewell-definedonlymoduloaninteger.αβ∂ai(k)=−ihα,k||β,ki,∂ki3.Dimensionalreductiontothethree-dimensionalZ2wherei,j,k,ℓ=1,2,3,4≡x,y,z,w).topologicalinsulatorUnlikethe(2+1)Dcase,theCStermislessrelevantthanthenon-topologicalMaxwelltermin(4+1)D,butisstillofThe4DgeneralizationoftheQHeffectgivesthefundamen-primaryimportancewhenunderstandingtopologicalphenom-talTRinvarianttopologicalinsulatorfromwhichalllower-enasuchasthechiralanomalyina(3+1)Dsystem,whichcandimensionaltopologicalinsulatorscanbederivedsystemati-beregardedastheboundaryofa(4+1)Dsystem(Qietal.,callybyaprocedurecalleddimensionalreduction(Qietal.,2008).Similartothe(2+1)DQHcase,thephysicalresponse2008).Startingfromthe(4+1)DCSfieldtheoryinof(4+1)DCSinsulatorsisgivenbyEq.(54),weconsiderfieldconfigurationswhereAµ(x)=µC2µνρστAµ(x0,x1,x2,x3)isindependentofthe“extradimension”j=2π2ǫ∂νAρ∂σAτ,(58)x4≡w,forµ=(x0,x1,x2,x3),andA4≡Awdependingonallcoordinates(x0,x1,x2,x3,x4).Weconsiderthege-whichisthenonlinearresponsetotheexternalfieldAµ.Toometrywherethe“extradimension”x4formsasmallcircle.understandthisresponsebetter,weconsideraspecialfieldInthiscase,thex4integralinEq.(54)canbecarriedoutex-configuration(Qietal.,2008):plicitly.AfterrestoringtheunitofelectronchargeeandfluxAx=0,Ay=Bzx,Az=−Ezt,Aw=At=0,(59)hc/efollowingtheconventioninelectrodynamics,weobtain 37aneffectiveTFTin(3+1)D:Zα3µνρτSθ=2dxdtθ(x,t)ǫFµνFρτ(x,t),(61)32πwhereα=e2/~c≃1/137isthefinestructureconstantandIθ(x,t)≡C2φ(x,t)=C2dx4A4(x,t,x4),(62)FIG.35Dimensionalreductionfrom(4+1)Dto(3+1)D.Thewhichcanbeinterpretedasthefluxduetothegaugefieldx4directioniscompactifiedintoasmallcircle,withafinitefluxφA4(x,t,x4)throughthecompactextradimension[Fig.35].threadingthroughthecircleduetothegaugefieldA4.TopreserveThefieldθ(x,t)iscalledtheaxionfieldinthefieldtheorylit-TRsymmetry,thetotalfluxcanbeeither0orπ,resultinginaZ2erature(Wilczek,1987).Inordertopreservethespatialandclassificationof3Dtopologicalinsulators.temporaltranslationsymmetry,θ(x,t)canbechosenasacon-stantparameterratherthanafield.Furthermore,wealreadyexplainedthattheoriginal(4+1)DCSTFTisTRinvariant.momentum-spaceisanalogoustothereal-spaceWZWterm.Therefore,itisnaturaltoaskhowcanTRsymmetrybepre-SimilartotheWZWterm,P3isonlywell-definedmoduloanservedinthedimensionalreduction.IfwechooseC2=1,integer,andcanonlytakethequantizedvaluesof0or1/2thenθ=φisthemagneticfluxthreadingthecompactifiedmoduloanintegerforanTRinvariantinsulator(Wangetal.,circle,andthephysicsshouldbeinvariantunderashiftofθ2010).by2π.TRtransformsθto−θ.Therefore,therearetwoandEssentialforthedefinitionofP3inEq.(63)istheTRin-onlytwovaluesofθwhichareconsistentwithTRsymmetry,varianceidentity(Wangetal.,2010):namelyθ=0andθ=π.Inthelattercase,TRtransformsG(k,−k)=TG(k,k)TT−1,(64)θ=πtoθ=−π,whichisequivalenttoθ=πmod2π.We00thereforeconcludethattherearetwodifferentclassesofTRwhichiscrucialforthequantizationofP3.Therefore,weinvarianttopologicalinsulatorsin3D,thetopologicallytrivialseethatunliketheinteger-classCSinsulators,theZ2insula-classwithθ=0,andthetopologicallynontrivialclasswithtorsaresymmetry-protectedtopologicalinsulators.Thiscanθ=π.beclearlyseenfromtheviewpointofthetopologicalorderAsjustseen,itismostnaturaltoviewthe3Dtopologicalparameter.Infact,thequantizationofP3isprotectedbyTRinsulatorasadimensionallyreducedversionofthe4Dtopo-symmetry.Inotherwords,ifTRsymmetryinEq.(64)isbro-logicalinsulator.However,formostphysicalsystemsin3D,ken,thenP3canbetunedcontinuously,andcanbeadiabati-wearegenerallygivenaninteractingHamiltonian,andwouldcallyconnectedfrom1/2to0.Thisisfundamentallydiffer-liketodefineatopologicalorderparameterthatcanbeeval-entfromtheCSinsulators,forwhichthecoefficientoftheuateddirectlyforany3DmodelHamiltonian.SincetheθCStermgivenbyEq.(55)isalwaysquantizedtobeaninte-anglecanonlytakethetwovalues0andπinthepresenceofger,regardlessofthepresenceorabsenceofsymmetries.ThisTRsymmetry,itcanbenaturallydefinedasthetopologicalinteger,ifnonzero,cannotbesmoothlyconnectedtozeropro-orderparameteritself.Foragenerallyinteractingsystem,itisvidedthattheenergygapremainsopen.Thereexistsamoregivenby(Wangetal.,2010):exhaustiveclassificationschemefortopologicalinsulatorsinθπZ1Zd4kvariousdimensions(Kitaev,2009;Qietal.,2008;Ryuetal.,P≡=duTrǫµνρσ[G∂G−1G∂G−132π6(2π)4µν2010)whichtakesintoaccounttheconstraintsimposedby0varioussymmetries.×G∂G−1G∂G−1G∂G−1],(63)ρσuForanoninteractingsystem,thefullGreen'sfunctionGinwherethemomentumk=(k1,k2,k3)isintegratedoverthetheexpressionforP3[Eq.(63)]isreplacedbythenoninteract-3DBrillouinzoneandthefrequencyk0isintegratedoveringGreen'sfunctionG0.Furthermore,thefrequencyintegral(−∞,+∞).G(k,u=0)≡G(k0,k,u=0)≡G(k0,k)canbecarriedoutexplicitly.Aftersomemanipulations,oneistheimaginary-timesingle-particleGreen'sfunctionofthefindsasimpleandbeautifulformulafullyinteractingmany-bodysystem,andG(k,u)foru6=0Z13ijk2isasmoothextensionofG(k,u=0),withafixedreferenceP3=2dkǫTr{[fij(k)−iai(k)aj(k)]ak(k(65))},16π3valueG(k,u=1)correspondingtotheGreen'sfunctionofatopologicallytrivialinsulatingstate.G(k,u=1)canbecho-whichexpressesP3astheintegraloftheCSformoverthe3DsenasadiagonalmatrixwithG=(ik−∆)−1foremptyαα0momentumspace.Forexplicitmodelsoftopologicalinsula-bandsαandG=(ik+∆)−1forfilledbandsβ,whereββ0tors,suchasthemodelbyZhangetal.(Zhangetal.,2009)∆>0isindependentofk.EventhoughP3isaphysicaldiscussedinSec.III.A,onecanevaluatethisformulaexplic-quantityin3D,aWZW(Witten,1983)typeofextensionpa-itlytoobtainrameteruisintroducedinitsdefinition,whichplaystheroleofk4intheformula(55)definingthe4DTI.ThedefinitioninP3=θ/2π=1/2(66) 38inthetopologicallynontrivialstate(Qietal.,2008).Es-inphasespacecanbealsobeperformed(Qietal.,2008).Thesin,MooreandVanderbiltalsocalculatedP3foravarietyphysicalconsequencesEq.(69)canbeunderstoodbystudy-ofinterestingmodels(Essinetal.,2009).Inagenericsys-ingthefollowingtwocases.temwithouttime-reversalorinversionsymmetry,theremay1.Half-QHeffectonthesurfaceofa3Dtopologicalinsula-benon-topologicalcontributionstotheeffectiveaction(61)tor.ConsiderasysteminwhichP3=P3(z)onlydependswhichmodifiestheformulaofP3giveninEq.(65).How-onz.Forexample,thiscanberealizedbythelatticeDiracever,suchcorrectionsvanisheswhentime-reversalsymme-model(Qietal.,2008)withθ=θ(z)(Fradkinetal.,1986;tryorinversionsymmetryispresent,sothatthequantizedWilczek,1987).Inthiscase,Eq.(69)becomesvalueP3=1/2(mod1)intopologicalinsulatorremainsro-µ∂zP3µνρbust(Essinetal.,2010;Malashevichetal.,2010).j=ǫ∂νAρ,µ,ν,ρ=t,x,y,2πSimilartothecaseof(4+1)DCSinsulators,thereisalsoanimportantdifferencebetweentheθtermfor(3+1)Dtopologi-whichdescribesaQHeffectinthexyplanewithHallcon-calinsulatorsandthe(2+1)DCStermforQHsystems,whichductivityσxy=∂zP3/2π.AuniformelectricfieldExalongweshallbrieflydiscuss(Maciejkoetal.,2010).In(2+1)D,thexdirectioninducesacurrentdensityalongtheydirec-thetopologicalCStermdominatesoverthenon-topologicaltionjy=(∂zP3/2π)Ex,theintegrationofwhichalongthezMaxwelltermatlowenergiesintherenormalizationgroupdirectiongivestheHallcurrentflow,asasimpleresultofdimensionalanalysis.However,inZz2Zz22D1(3+1)D,theθtermhasthesamescalingdimensionastheJy=dzjy=dP3Ex,Maxwellterm,andisthereforeequallyimportantatlowener-z12πz1gies.ThefullsetofmodifiedMaxwell'sequationsincludingwhichcorrespondstoa2DQHconductancethetopologicalterm(62)isgivenbyZz2σ2D=dP/2π.(70)1µνµναµνστ1µxy3∂νF+∂νP+ǫ∂ν(P3Fστ)=j,(67)z14π4πcForainterfacebetweenatopologicallynontrivialinsulatorwhichcanbewrittenincomponentformaswithP3=1/2andatopologicallytrivialinsulatorwith∇·D=4πρ+2α(∇P3·B),P3=0,whichcanbetakenasthevacuum,theHallconduc-1∂D4π1tanceisσH=∆P3=±1/2.Asidefromanintegerambi-∇×H−c∂t=cj−2α(∇P3×E)+c(∂tP3)B,guity,theQHconductanceisexactlyquantized,independentofthedetailsoftheinterface.AsdiscussedinsectionIII.D.1,1∂B∇×E+=0,thehalfquantumHalleffectonthesurfaceisareflectionofc∂tthebulktopologywithP3=1/2,andcannotbedetermined∇·B=0,(68)purelyfromthelowenergysurfacemodels.whereD=E+4πPandH=B−4πMonlyincludethe2.Topologicalmagnetoelectriceffectinducedbyatemporalnon-topologicalcontributions.Alternatively,onecanusethegradientofP3.Havingconsideredatime-independentP3,weordinaryMaxwell'sequationswithmodifiedconstituentequa-nowconsiderthecasewhenP3=P3(t)isspatiallyuniform,tions(40).ThesesetofmodifiedMaxwellequationsarecalledbuttime-dependent.Equation(69)nowbecomestheaxionelectrodynamicsinfieldtheory(Wilczek,1987).EventhoughtheconventionalMaxwelltermandthetopo-i∂tP3ijkj=−ǫ∂jAk,i,j,k=x,y,z,2πlogicaltermarebothpresent,thereexistexperimentaldesignswhichcaninprincipleextractthepurelytopologicalcontri-whichcanbesimplywrittenasbutions(Maciejkoetal.,2010).Furthermore,thetopological∂tP3responseiscompletelycapturedbytheTFT,whichweshallj=−B.(71)2πdiscuss.StartingfromtheTFT[Eq.61],wetakeafunctionalderivativewithrespecttoAµ,andobtainthecurrentasBecausethechargepolarizationPsatisfiesj=∂tP,wecanintegrateEq.71inastatic,uniformmagneticfieldBtogetµ1µνστ∂P=−∂(PB/2π),sothatj=ǫ∂νP3∂σAτ,(69)tt32πBwhichisthegeneraltopologicalresponseof(3+1)Dinsula-P=−(P3+const.).(72)tors.ItisworthnotingthatwedonotassumeTRinvariance2πhere,otherwiseP3shouldbequantizedtobeintegerorhalf-Thisequationdescribesthechargepolarizationinducedbyainteger.Infact,hereweassumeainhomogeneousP3(x,t).magneticfield,whichisamagnetoelectriceffect.Thepromi-Itisinterestingtonoticethatthiselectromagneticresponsenentfeaturehereisthatitisexactlyquantizedtoahalf-integerlooksverysimilartothe4DresponseinEq.58,withtheonlyforaTRinvarianttopologicalinsulator,whichiscalledthedifferencethatA4isreplacedbyP3inEq.(69).Thisisaman-topologicalmagnetoelectriceffect(Qietal.,2008)(TME).ifestationofdimensionalreductionattheleveloftheelectro-AnotherrelatedeffectoriginatingfromtheTFTistheWit-magneticresponse.Amoresystematictreatmentonthistopicteneffect(Qietal.,2008;Witten,1979).Forthisdiscussion, 39weassumethattherearemagneticmonopoles.Forauniform5.GeneralphasediagramoftopologicalMottinsulatorandP3,Eq.(71)leadstotopologicalAndersoninsulator∂tP3∇·j=−∇·B.(73)Sofar,wehaveintroducedtopologicalorderparam-2πetersforTRinvarianttopologicalinsulatorsin4D,3DEvenifmagneticmonopoledoesnotexistaselementarypar-and2D.Thesetopologicalorderparametersaredefinedinticles,foralatticesystem,themonopoledensityρm=∇·termsofthefullsingle-particleGreen'sfunction.Acau-B/~2πcanstillbenonvanishing,andweobtaintioninorderisthatthesetopologicalorderparametersarenotapplicabletofractionalstateswithgroundstatede-∂tρe=(∂tP3)ρm.(74)generacy(BernevigandZhang,2006;LevinandStern,2009;Therefore,whenP3isadiabaticallychangedfromzerotoMaciejkoetal.,2010;Swingleetal.,2010),forwhichaTFTΘ/2π,themagneticmonopolewillacquireanelectricchargeapproachisstillpossible,butsimpletopologicalorderpa-oframetersarehardertofind.In3D,fractionaltopologicalin-sulatorsarecharacterizedbyatopologicalorderparameterΘQe=Qm,(75)P3thatisarationalmultipleof1/2(Maciejkoetal.,2010;2πSwingleetal.,2010).SuchstatesareconsistentwithTRwhereQmisthemagneticcharge.SucharelationwassymmetryiffractionallychargedexcitationsandgroundstatefirstderivedbyWitteninthecontextofthetopologicaldegeneraciesonspatialmanifoldsofnontrivialtopologyareterminquantumchromodynamics(Witten,1979),andlaterpresent.WhenTRsymmetryisbrokenonthesurface,afrac-discussedinthecontextoftopologicalinsulators(Qietal.,tionalTMEgivesrisetohalfofafractionalQHeffectonthe2008;RosenbergandFranz,2010).Thiseffectcouldalsoap-surface(Maciejkoetal.,2010;Swingleetal.,2010).pearunderadifferentguiseintopologicalexcitoncondensa-Nextweshalldiscussthephysicalconsequencesimpliedtion(Seradjehetal.,2009),whereae/2chargeisinducedbybythetopologicalorderparametersuchasP2andP3.Theavortexintheexcitoncondensate,whichservesasthe“mag-discussionweshallpresentisverygeneralanditsapplica-neticmonopole”.bilitydoesnotdependonthespatialdimensions.Further-more,sincethetopologicalorderparametersareexpressedintermsofthefullGreen'sfunctionofaninteractingsys-4.Furtherdimensionalreductiontothetwo-dimensionalZ2topologicalinsulatortem,theycanbeusefultogeneralinteractingsystems.Sup-posewehaveafamilyofHamiltonianslabeledbyseveralNowweturnourattentionto(2+1)DTRinvariantZ2parameters.Tobespecific,weconsideratypicalphasedia-topologicalinsulators.SimilartotheWZW-typetopologicalgram(Wangetal.,2010)[Fig.(36)]foraninteractingHamil-orderparameterP3in(3+1)D,thereisalsoatopologicaltonianH=H0(λ)+H1(g),whereH0isthenoninter-†orderparameterdefinedfor(2+1)DTRinvariantinsulators.actingpartincludingtermssuchastijcicj,andH1istheThemaindifferencebetween(2+1)Dand(3+1)Disthatinelectron-electroninteractionpartincludingtermssuchasthe(2+1)DweneedtwoWZWextensionparametersuandv,inHubbardinteractiongni↓ni↑.Thesetwopartsaredeterminedcontrasttoasingleparameteruin(3+1)D,whichisaman-bysingle-particleparametersλ=(λ1,λ2,···)andcouplingifestationofthefactthatbothdescendfromthefundamentalconstantsg=(g1,g2,···).When(λ,g)aresmoothlytuned,(4+1)Dtopologicalinsulator(ZhangandHu,2001).Forathegroundstatealsoevolvessmoothly,aslongastheenergygeneralinteractinginsulator,the(2+1)Dtopologicalordergapremainsopenandthetopologicalorderparameterssuchasparameterisexpressedas(Wangetal.,2010)P2andP3remainunchanged.OnlywhenthegapclosesandZ1Z1Z3thefullGreen'sfunctionGbecomessingular,thesetopolog-1µνρστdk−1icalorderparametershaveajump,asindicatedbythecurveP2=ǫdudv3Tr[G∂µG120−1−1(2π)abinFig.(36).ThemostimportantpointinFig.36canbeil-×G∂G−1G∂G−1G∂G−1G∂G−1lustratedbyconsideringtheverticallineBF.StartingfromaνρστnoninteractingstateF,oneadiabaticallytunesoninteractions,=0or1/2(modZ),(76)andeventuallythereisaphasetransitionatEtothetopolog-whereǫµνρστisthetotallyantisymmetrictensorinfivedi-icalinsulator(TI)state.TheinteractingstateB,whichisanmensions,takingvalue1whenthevariablesareaneveninteraction-inducedtopologicalinsulatorstate,hasadifferentpermutationof(k0,k1,k2,u,v).ThecasesP2=0andtopologicalorderparameterfromthecorrespondingnoninter-P2=1/2moduloanintegercorrespondtotopologicallytriv-actingnormalinsulator(NI)stateF.ThedifferenceintheialandnontrivialTRinvariantinsulatorsin(2+1)D,respec-topologicalorderparameterthusprovidesacriterionfordis-tively.Thistopologicalorderparameterisvalidforinteract-tinguishingtopologicalclassesofinsulatorsinthepresenceofingQSHsystemsin(2+1)D,includingstatesintheMottgeneralinteractions.regime(Raghuetal.,2008).P2canbephysicallymeasuredTherehavealreadybeenseveraltheoreticalproposalsofbythefractionalchargeattheedgeoftheQSHstate(Qietal.,stronglyinteractingtopologicalinsulators,i.e.topological2008).Mottinsulators(Raghuetal.,2008).2Dtopologicallynon- 40B.TopologicalbandtheoryWeshallnowgiveabriefintroductiontoTBT.Eventhoughthistheoryisonlyvalidfornon-interactingsystems,ithasbecomeanimportanttoolinthediscoveryofnewtopo-logicalmaterials.Unfortunately,evaluatingtheZ2invari-antsforagenericbandstructureisingeneraladifficultproblem.Severalapproacheshavebeenexploredinthelit-eratureincludingspinChernnumbers(FukuiandHatsugai,FIG.36Phasediagraminthe(λ,g)plane.Thedarkcurveabis2007;Prodan,2009;Shengetal.,2006),topologicalinvari-thephaseboundaryseparatingnormalinsulators(NI)andtopolog-antsconstructedfromBlochwavefunctions(FuandKane,icalinsulators(TI).Allphasesaregapped,exceptonab.Thetrue2006;KaneandMele,2005;MooreandBalents,2007;Roy,parameterspaceisinfactinfinitedimensional,butthis2Ddiagram2009),anddiscreteindicescalculatedfromsingle-particleillustratesthemainfeatures.FromWangetal.,2010.statesatTRIMintheBrillouinzone(FuandKane,2007).Wewillfocusonthelastmethodforitssimplicity(FuandKane,2007).Thisbasicquantityinthisapproachisthematrixelementoftrivialinsulatingstateshavebeenobtainedfromthecom-theTRoperatorTbetweenstateswithTRconjugatemomentabinationofatrivialnoninteractingbandstructureandin-kand−k(FuandKane,2006),teractionterms(GuoandFranz,2009;Raghuetal.,2008;WeeksandFranz,2010).SuchtopologicalinsulatorscanbeBαβ(k)=h−k,α|T|k,βi.(77)regardedastopologicallynontrivialstatesarisingfromdy-namicallygeneratedSOC(Wuetal.,2007;WuandZhang,SinceBαβisdefinedasamatrixelementbetweenBlochstates2004).TheeffectofinteractionsonthetheQSHstatehasalsoatTRconjugatemomenta,itisexpectedthatthisquantitybeenrecentlystudied(RachelandHur,2010).In3D,strongcontainssomeinformationaboutthebandtopologyofTRtopologicalinsulatorswithtopologicalexcitationshavebeeninvarianttopologicalinsulators.AttheTRIMΓi,B(k=obtained(PesinandBalents,2010;Zhangetal.,2009).Topo-Γi)isantisymmetric,sothatthefollowingquantitycanbelogicalinsulatorshavebeensuggestedtoexistintransitiondefined(FuandKane,2006):metaloxides(Shitadeetal.,2009),wherethecorrelationef-pfectisstrong.Itwasalsoproposedthatonecouldachievethedet[B(Γi)]δi=.(78)topologicalinsulatorstateinKondoinsulators(Dzeroetal.,Pf[B(Γi)]2010).AllthesetopologicalMottinsulatorstatescanbeun-inwhichPfstandsforthePfaffianofanantisymmetricma-derstoodintheframeworkofthetopologicalorderparametertrix.SincePf[B(Γ)]2=det[B(Γ)],wehaveδ=±1.ItexpressedintermsofthefullGreen'sfunction(Wangetal.,iiishouldbenoticedthatthewavefunctions|k,αimustbecho-2010).Interaction-inducedtopologicalinsulatorstatessuchassencontinuouslyinBZtoavoidambiguityinthedefinitionthetopologicalMottinsulatorsproposedinRef.(Raghuetal.,ofδi.In1D,thereareonlytwoTRIM,anda“TRpolariza-2008)correspondtoregionsrepresentedbythepointBintion”(FuandKane,2006)canbedefinedastheproductofFig.36,whichhasatrivialnoninteractingunperturbedHamil-δi,tonianH0(B)butacquiresanontrivialtopologicalorderpa-rameterduetotheinteractionpartH1(B)oftheHamilto-π≡(−1)Pθ=δδ,(79)12nian.ThepreviouslydiscussedtopologicalorderparametersareusefulfordeterminingthephasediagramsofinteractingwhichisaZ2analogtothechargepolariza-systems.tion(King-SmithandVanderbilt,1993;Resta,1994;Thouless,1983;Zak,1989).AfurtheranalogybetweenFordisorderedsystems,thetopologicalorderparametersthechargepolarizationandtheTRpolarizationsuggestsarestillapplicable,providedthatweusethedisorder-averagedtheformoftheZ2invariantforTRinvarianttopologicalGreen'sfunctions.Inthiscase,Fig.36canberegardedasinsulators.Ifanangularparameterθistunedfrom0to2π,asimplephasediagramofdisorderedsystems,withginter-thechangeinthechargepolarizationPaftersuchacycleispretedasthedisorderstrength.TherepresentativepointBisexpressedasthefirstChernnumberC1inthe(k,θ)space.Inadisorder-inducedtopologicalinsulatorstate.Thedisorder-fact,thesameC1wouldgivestheTKNNinvariant(Thouless,inducedTIstatehasbeenstudiedrecently(Grothetal.,1983)ifθwereregardedasamomentum.Byanalogywith2009;Guo,2010;Guoetal.,2010;Imuraetal.,2009;P,theZ2invariantfor(2+1)DtopologicalinsulatorscanbeJiangetal.,2009;Lietal.,2009;LoringandHastings,2010;definedasObuseetal.,2008;Olshanetskyetal.,2010;Ostrovskyetal.,2009;ShindouandMurakami,2009).Therefore,thetopolog-(−1)ν2D=(−1)Pθ(k2=0)−Pθ(k2=π),(80)icalorderparameterspreviouslydiscussedhavetheabilitytodescribebothinteractinganddisorderedsystems.wherePθ(k2)=δ1δ2,δiisdefinedattheTRIMk1=0orπ, 41kxkxE(a)E(b)(a)kk(b)xxL2L2eG21G22Lb3kk4ezzeL1L1a2eb2GGkea21112yL3b1ekyb1eea1a1kkFIG.37(a)The2DbulkBrillouinzoneprojectedontothe1DedgeLLLLBrillouinzone.ThetwoedgeTRIMΛabab1andΛ2areprojectionsofpairsofthefourbulkTRIMΓi=(aµ).(b)ProjectionoftheTRIMofthe3DBrillouinzoneontoa2DsurfaceBrillouinzone.FromFIG.38SchematicrepresentationofthesurfaceenergylevelsofaFuandKane,2007.crystalineither2Dor3D,asafunctionofsurfacecrystalmomentumonapathconnectingTRIMΛaandΛb.Theshadedregionshowsthebulkcontinuumstates,andthelinesshowdiscretesurface(oredge)andk2isregardedasaparameter.ExpandingEq.(80)givesbandslocalizednearoneofthesurfaces.TheKramersdegenerateY4surfacestatesatΛaandΛbcanbeconnectedtoeachotherintwo(−1)ν2D=δ,(81)possibleways,shownin(a)and(b),whichreflectthechangeinTRipolarization∆Pθofthecylinderbetweenthosepoints.Case(a)oc-i=1cursintopologicalinsulators,andguaranteesthesurfacebandscrosswherei=1,2,3,4labelsthefourTRIMinthe2DBril-anyFermienergyinsidethebulkgap.FromFuetal.,2007.louinzone.(−1)ν2D=+1impliesatrivialinsulatorwhile(−1)ν2D=−1impliesatopologicalinsulator.Furthermore,asaTRpolarization,ν2DalsodeterminesthewayinwhichWhenν0=0,statesareclassifiedaccordingtoGν,andareKramerspairsofsurfacestatesareconnected[Fig.(38)],calledweaktopologicalinsulators(Fuetal.,2007)whenthewhichsuggeststhatbulktopologyandedgephysicsareinti-weakindicesνkareodd.matelyrelated.Thisisanotherexampleofthe“holographicHeuristicallythesestatescanbeinterpretedasstackedQSHprinciple”fortopologicalphenomenaincondensedmatterstates.Asanexample,considerplanesofQSHstackedinthephysics.zdirection.Whenthecouplingbetweenthelayersiszero,theWenowdiscuss3Dtopologicalinsulators.Itisinterestingbanddispersionwillbeindependentofkz.ItfollowsthatthetonotethatinTBTthenaturalrouteis“dimensionalincrease”,fourδi'sassociatedwiththeplanekz=π/awillhaveprod-incontrasttothe“dimensionalreduction”procedureofTFT.uct−1andwillbethesameasthefourassociatedwiththeFromthisdimensionalincrease,the3D(strong)topologicalplanekz=0.Thetopologicalinvariantswillthenbegiveninvariantisnaturallydefinedas(FuandKane,2007;Fuetal.,byν0=0andGν=(2π/a)ˆz.Thisstructurewillremain2007)whenhoppingbetweenthelayersisintroduced.Moregener-ally,whenQSHstatesarestackedintheG,directionthein-Y8ν3DvariantwillbeGν=Gmod2.ThisimpliesthatQSHstates(−1)=δi.(82)stackedalongdifferentdirectionsG1andG2areequivalenti=1ifG1=G2mod2(FuandKane,2007).Asforthesur-Inadditiontothestronginvariant,ithasbeenshownthatfacestates,whenthecouplingbetweenthelayersiszero,itistheproductofanyfourδi'sforwhichtheΓilieinthesameclearthatthegapinthe2Dsystemimpliestherewillbenoplaneisalsogaugeinvariant,anddefinestopologicalinvari-surfacestatesonthetopandbottomsurfaces;onlythesideantscharacterizingthebandstructure(FuandKane,2007).surfaceswillhavegaplessstates.WecanalsothinkabouttheThisfactleadstothedefinitionofthreeadditionalinvariantsinstabilityofthesurfacestatesfortheweakinsulators.Infact,3Dknownasweaktopologicalinvariants(FuandKane,2007;weaktopologicalinsulatorsareunstablewithrespecttodis-Fuetal.,2007;MooreandBalents,2007;Roy,2009).Theseorder.WecanheuristicallyseethattheyarelessstablethanZ2invariantscanbearrangedasa3Dvectorwithelementsνkthestronginsulatorsinthefollowingway.IfwestackanoddgivenbynumberofQSHlayers,therewouldatleastbeonedelocalizedYsurfacebranch.However,thesurfacestatesforanevennum-(−1)νk=δ,(83)i=(n1n2n3)beroflayerscanbecompletelylocalizedbydisorderorpertur-nk=1;nj6=k=0,1bations.Despitethisinstability,ithasbeenshown(Ranetal.,2009)thattheweaktopologicalinvariantsguaranteetheexis-where(ν1ν2ν3)dependonthechoiceofreciprocallatticevec-tenceofgaplessmodesoncertaincrystaldefects.Foradislo-torsandareonlystrictlywelldefinedwhenawell-definedlat-cationwithBurgersvectorbitwasshownthattherewillbeticeispresent.Itisusefultoviewtheseinvariantsascompo-nentsofamod2reciprocallatticevector,gaplessmodesonthedislocationifGν·b=(2n+1)πforintegern.Gν=ν1b1+ν2b2+ν3b3.(84)Similartothe2Dtopologicalinsulator,thereareconnec- 420;(001)0;(011)0;(111)1;(111)C.Reductionfromtopologicalfieldtheorytotopologicalkykykykybandtheory+-+-+-+++-+++++-WenowbrieflydiscusstherelationbetweentheTFTand++kx++kx++kx++kxTBT.Ontheonehand,theTFTapproachisverypowerfulto+++--+++kkkkrevealvariousaspectsofthelow-energyphysics,anditalsozzzzkkkkyyyyprovideadeepunderstandingoftheuniversalityamongdif-kkkkferentsystems.Furthermore,incontrasttoTBT,TFTisvalidxxxxforinteractingsystems.Ontheotherhand,fromapracticalviewpoint,wealsoneedfastalgorithmstocalculatetopolog-icalinvariants,whichisthegoalofTBT.Anintuitiveunder-standingoftheTBTofZ2topologicalinsulatorsisasfollows.FIG.39DiagramsdepictingfourdifferentphasesindexedbyForinteger-classCStopologicalinsulators,thetopologicalin-ν0;(ν1ν2ν3).ThetoppaneldepictsthesignsofδiatthepointsΓivariantCnisexpressedastheintegralofGreen'sfunctionsontheverticesofacube.Thebottompanelcharacterizesthebandstructureofa001surfaceforeachphase.Thesolidandopencircles(orBerrycurvature,inthenoninteractinglimit).Therefore,depicttheTRpolarizationπaatthesurfacemomentaΛa,whicharetheknowledgeofBlochstatesoverthewholeBrillouinzoneprojectionsofpairsofΓiwhichdifferonlyintheirzcomponent.isneededtocalculateCn.ForTRinvariantZ2topologicalThethicklinesindicatepossibleFermiarcswhichenclosespecificinsulators,theTRsymmetryconstraintenablesustodeter-Λa.FromFuetal.,2007.minethetopologicalclassofagiveninsulatorwithlessin-formation:wedonotneedtheinformationovertheentireBrillouinzone.Forinsulatorswithinversionsymmetry,thetionsbetweenthebulkinvariantsof3Dtopologicalinsulatorparityatseveralhigh-symmetrypointscompletelydeterminesandthecorresponding2Dsurfacestatespectrum.Asasam-thetopologicalclass(FuandKane,2007),whichexplainstheple,Fig.(39)showsfourdifferenttopologicalclassesfor3DsuccessofTBT.Asnaturallyexpected,theTBTapproachisbandstructureslabeledwiththecorresponding(ν0;ν1ν2ν3).relatedtotheTFTapproach.Infact,ithasbeenrecentlyTheeightΓiarerepresentedastheverticesofacubeinmo-proved(Wangetal.,2010)thattheTFTdescriptioncanbementumspace,withthecorrespondingδishownas±signs.exactlyreducedtotheTBTinthenoninteractinglimit.WeThelowerpanelshowsacharacteristicsurfaceBrillouinzonenowoutlinethisproof(Wangetal.,2010).Startingfromthefora001surfacewiththefourΛalabeledbyeitherfilledorexpressionforP3inEq.(63)andEq.(65),onecanshowthatsolidcircles,dependingonthevalueofπa=δi=(a1)δi=(a2).ZGenericallyitisexpectedthatthesurfacebandstructurewill13ijk††resembleFig.38(b)onpathsconnectingtwofilledcirclesor2P3(mod2)=−2dkǫTr[(B∂iB)(B∂jB)24πtwoemptycircles,andwillresembleFig.38(a)onpathscon-×(B∂B†)](mod2).(86)knectingafilledcircletoanemptycircle(FuandKane,2007).Thisconsiderationdeterminesthe2Dsurfacestatesqualita-Bysometopologicalargument,thisexpressionfortively.P3isshowntogivethedegreedegfofcertainIfaninsulatorhasinversionsymmetry,thereisasimplemap(Dubrovinetal.,1985)fromtheBrillouinzonethree-torusT3totheSU(2)groupmanifold.TherearetwoalgorithmtocalculatetheZ2invariant(FuandKane,2007):indeed,thereplacementinEq.(81)andEq.(82)ofδibyseeminglydifferentexpressionsfordegf,oneofwhichisofintegralformasgivenbyEq.(86),whiletheotherisYNofdiscreteformandgivensimplybythenumberofpointsδi=ξ2m(Γi),(85)mappedtoaarbitrarilychosenimageinSU(2).DuetoTRm=1symmetry,ifwechoosetheimagepointasoneofthetwogivesthecorrectZ2invariants.Hereξ2m(Γi)=±1istheantisymmetricmatricesinSU(2)(e.g.iσy),wehaveanparityeigenvalueofthe2mthoccupiedenergybandatΓiinteresting“pairannihilation”ofthosepointsotherthanthe[Fig.37],whichsharesthesameeigenvalueξ2m=ξ2m−1eightTRIM(Wangetal.,2010).ThefinalresultisexactlywithitsdegenerateKramerspartner(FuandKane,2007).ThetheZ2invariantfromTBT.TheexplicitrelationbetweenTFTproductisonlyoverhalfoftheoccupiedbands.SincetheandTBTis(Wangetal.,2010)definitionoftheδireliesonparityeigenvalues,theδiare2P3ν3D(−1)=(−1).(87)onlywell-definedinthiscasewheninversionsymmetryispresent(FuandKane,2007).However,forinsulatorswith-outinversionsymmetry,thisalgorithmisveryuseful.Infact,V.TOPOLOGICALSUPERCONDUCTORSANDifwecandeformagiveninsulatortoaninversion-symmetricSUPERFLUIDSinsulatorandkeeptheenergygapopenalongtheway,theresultantZ2invariantsarethesameastheinitialonesdueSoonaftertheirdiscovery,thestudyofTRinvarianttopo-totopologicalinvariance,butcanbecalculatedfrompar-logicalinsulatorswasgeneralizedtoTRinvarianttopologi-ity(FuandKane,2007).calsuperconductorsandsuperfluids(Kitaev,2009;Qietal., 432009;Roy,2008;Schnyderetal.,2008).Thereisadirectphysicaldegreesoffreedom,eachchiralMajoranaedgestateanalogybetweensuperconductorsandinsulatorsbecausethehashalfthedegreesoffreedomofthechiraledgestateofaBogoliubov-deGennes(BdG)Hamiltonianforthequasipar-QHsystem.Therefore,thechiralsuperconductoristhe“min-ticlesofasuperconductorisanalogoustotheHamiltonianofimal”topologicalstatein2D.Theanalogybetweenachiralabandinsulator,withthesuperconductinggapcorrespondingsuperconductorandaQHstateisillustratedintheupperpan-tothebandgapoftheinsulator.elsofFig.40.Followingthesameanalogy,onecanconsider3He-Bisanexampleofsuchatopologicalsuperfluidstate.thesuperconductinganalogofQSHstate—a“helical”su-ThisTRinvariantstatehasafullpairinggapinthebulk,perconductorinwhichfermionswithupspinsarepairedinandgaplesssurfacestatesconsistingofasingleMajoranathepx+ipystate,andfermionswithdownspinsarepairedcone(ChungandZhang,2009;Qietal.,2009;Roy,2008;inthepx−ipystate.SuchaTRinvariantstatehasafullgapSchnyderetal.,2008).Infact,theBdGHamiltonianforinthebulk,andcounter-propagatinghelicalMajoranastates3He-BisidenticaltothemodelHamiltonianofa3Dtopo-attheedge.Incontrast,theedgestatesoftheTRinvariantlogicalinsulator(Zhangetal.,2009),andinvestigatedexten-topologicalinsulatorarehelicalDiracfermionswithtwicesivelyinSec.III.A.In2D,theclassificationoftopologicalthedegreesoffreedom.AsisthecasefortheQSHstate,asuperconductorsisverysimilartothatoftopologicalinsula-masstermfortheedgestatesisforbiddenbyTRsymmetry.tors.TRbreakingsuperconductorsareclassifiedbyaninte-Therefore,suchasuperconductingphaseistopologicallypro-ger(ReadandGreen,2000;Volovik,1988),similartoquan-tectedinthepresenceofTRsymmetry,andcanbedescribedtumHallinsulators(Thoulessetal.,1982),whileTRinvariantbyaZ2topologicalquantumnumber(Kitaev,2009;Qietal.,superconductorsareclassified(Kitaev,2009;Qietal.,2009;2009;Roy,2008;Schnyderetal.,2008).Thefourtypesof2DRoy,2008;Schnyderetal.,2008)byaZ2invariantin1Dtopologicalstatesofmatterdiscussedherearesummarizedinand2D,butbyaninteger(Z)invariantin3D(Kitaev,2009;Fig.40.Schnyderetal.,2008).Asastartingpoint,wefirstconsidertheHamiltonianoftheBesidestheTRinvarianttopologicalsuperconductors,simplestnontrivialTRbreakingsuperconductor,thep+ipsu-theTRbreakingtopologicalsuperconductorshavealsoat-perconductor(ReadandGreen,2000)forspinlessfermions:!!tractedalotofinterestrecently,becauseoftheirrela-X
tionwithnon-Abelianstatisticsandtheirpotentialapplica-1†ǫp∆p+cpH=2cp,c−p∗†,(88)tiontotopologicalquantumcomputation.TheTRbreak-p∆p−−ǫpc−pingtopologicalsuperconductorsaredescribedbyaninte-2withǫp=p/2m−µandp±=px±ipy.IntheweakgerN.Thevortexofatopologicalsuperconductorwithpairingphasewithµ>0,thepx+ipychiralsuperconductoroddtopologicalquantumnumberNcarriesanoddnum-isknowntohavechiralMajoranaedgestatespropagatingonberofMajoranazeromodes(Volovik,1999),givingrisetoeachboundary,describedbytheHamiltoniannon-Abelianstatistics(Ivanov,2001;ReadandGreen,2000)Xwhichcouldprovideaplatformfortopologicalquantumcom-Hedge=vFkyψ−kyψky,(89)puting(Nayaketal.,2008).ThesimplestmodelforanN=1ky≥0chiraltopologicalsuperconductorisrealizedinthepx+ipy†pairingstateofspinlessfermions(ReadandGreen,2000).Awhereψ−ky=ψkyisthequasiparticlecreationopera-spinfulversionofthechiralsuperconductorhasbeenpre-tor(ReadandGreen,2000)andtheboundaryistakenparalleldictedtoexistinSr2RuO4(MackenzieandMaeno,2003),buttotheydirection.Thestrongpairingphaseµ<0istriv-theexperimentalsituationisfarfromdefinitive.Recently,ial,andthetwophasesareseparatedbyatopologicalphaseseveralnewproposalstorealizeMajoranafermionstatesintransitionatµ=0.conventionalsuperconductorshavebeeninvestigatedbymak-IntheBHZmodelfortheQSHstateininguseofstrongSOC(FuandKane,2008;Qietal.,2010;HgTe(Bernevigetal.,2006),ifweignorethecouplingSauetal.,2010).termsbetweenspinupandspindownelectrons,thesystemisadirectproductoftwoindependentQHsystemsinwhichspinupandspindownelectronshaveoppositeHallconductance.Inthesameway,thesimplestmodelforthetopologicallyA.Effectivemodelsoftime-reversalinvariantsuperconductorsnontrivialTRinvariantsuperconductorin2DisgivenbythefollowingHamiltonian:ThesimplestwaytounderstandTRinvarianttopologicalǫp∆p+00superconductorsisthroughtheiranalogywithtopologicalin-1X∗sulators.The2Dchiralsuperconductingstateisthesupercon-H=Ψ˜†∆p−−ǫp00Ψ˜(90),200ǫp−∆∗p−ductoranalogoftheQHstate.AQHstatewithChernnum-pberNhasNchiraledgestates,whileachiralsuperconductor00−∆p+−ǫpwithtopologicalquantumnumberNhasNchiralMajoranaT††edgestates.SincethepositiveandnegativeenergystatesofwithΨ(˜p)≡c↑p,c↑−p,c↓p,c↓−p.FromEq.(90)wetheBdGHamiltonianofasuperconductordescribethesameseethatspinup(down)electronsformpx+ipy(px−ipy) 44(ψky↑,ψ−ky↓)transformsunderTRasaKramersdoublet,whichforbidsagapintheedgestatespectrumwhenTRispreservedbypreventingthemixingofspinupandspindownmodes.Toseethisexplicitly,noticethattheonlyky-independenttermthatcanbeaddedtotheedgeHamiltonianP(91)isimkyψ−ky↑ψky↓,withmreal.However,suchatermisoddunderTR,whichimpliesthatanybackscatteringbetweenquasiparticlesisforbiddenbyTRsymmetry.Thedis-cussionaboveisexactlyparalleltotheZ2topologicalcharac-terizationofQSHsystem.Infact,theHamiltonian(90)hasexactlythesameformasthefour-bandeffectiveHamiltonianFIG.40(Toprow)Schematiccomparisonof2Dchiralsuperconduc-oftheQSHeffectinHgTequantumwells(Bernevigetal.,torandQHstate.Inbothsystems,TRsymmetryisbrokenandthe2006).TheedgestatesoftheQSHinsulatorconsistofanedgestatescarryadefinitechirality.(Bottomrow)Schematiccom-oddnumberofKramerspairs,whichremaingaplessun-parisonof2DTRinvarianttopologicalsuperconductorandQSHin-deranysmallTRinvariantperturbation(Wuetal.,2006;sulator.BothsystemspreserveTRsymmetryandhaveahelicalpairXuandMoore,2006).Sucha“helicalliquid”withanoddofedgestates,whereoppositespinstatescounter-propagate.ThedashedlinesshowthattheedgestatesofthesuperconductorsareMa-numberofKramerspairsattheFermienergycannotbereal-joranafermionssothattheE<0partofthequasiparticlespectrumizedinanybulk1Dsystem,andcanonlyappearholograph-isredundant.Intermsoftheedgestatedegreesoffreedom,wehaveicallyastheedgetheoryofa2DQSHinsulator(Wuetal.,224symbolicallyQSH=(QH)=(HelicalSC)=(ChiralSC).2006).Similarly,theedgestatetheoryEq.(91)canbecalledFromQietal.,2009.a“helicalMajoranaliquid”,andcanonlyexistonthebound-aryofaZ2topologicalsuperconductor.Oncesuchatopolog-icalphaseisestablished,itisrobustunderanyTRinvariantCooperpairs,respectively.ComparingthismodelHamilto-perturbationssuchasRashba-typeSOCands-wavepairing,nian(90)forthetopologicalsuperconductorwiththeBHZevenifspinrotationsymmetryisbroken.TheedgehelicalmodeloftheHgTetopologicalinsulator[Eq.2],wefirstseeMajoranaliquidcanbedetectedbyelectrictransportthroughthatthetermproportionaltotheidentitymatrixintheBHZaquantumpointcontactbetweentwotopologicalsupercon-modelisabsenthere,reflectingthegenericparticle-holesym-ductors(Asanoetal.,2010).metryoftheBdGHamiltonianforsuperconductors.OntheaThe2DHamiltonian(90)describesaspin-tripletpairing,otherhand,thetermsproportionaltothePaulimatricesσarethespinpolarizationofwhichiscorrelatedwiththeorbitalidenticalinbothcases.Therefore,atopologicalsuperconduc-angularmomentumofthepair.Suchacorrelationcanbenat-torcanbeviewedasatopologicalinsulatorwithparticle-holeurallygeneralizedto3Dwherespinpolarizationandorbitalsymmetry.ThetopologicalsuperconductorHamiltonianalsoangularmomentumarebothvectors.TheHamiltonianofsuchhashalfasmanydegreesoffreedomasthetopologicalinsu-a3Dsuperconductorisgivenbylator.ThemodelHamiltonian(90)isexpressedintermsof!theNambuspinorΨ(˜p)whichartificiallydoublesthedegrees1X†ǫpI2×2iσ2σα∆αjpjH=ΨΨ,(92)offreedomascomparedtothetopologicalinsulatorHamilto-2ph.c.−ǫpI2×2nian.Bearingthesedifferencesinmind,inanalogywiththeQSHsystem,weknowthattheedgestatesoftheTRinvariantwhereweuseadifferentbasisΨ(p)≡TsystemdescribedbytheHamiltonian(90)consistofspinupc,c,c†,c†.∆αjisa3×3matrixwith↑p↓p↑−p↓−pandspindownquasiparticleswithoppositechiralities:α=1,2,3andj=x,y,z.Interestingly,anexampleofsuchaHamiltonianisgivenbythewell-known3He-Bphase,forX
αjHedge=vFkyψ−ky↑ψky↑−ψ−ky↓ψky↓.(91)whichtheorderparameter∆isdeterminedbyanorthogo-nalmatrix∆αj=∆uαj,u∈SO(3)(VollhardtandW¨olfle,ky≥01990).Hereandbelowweignorethedipole-dipoleinter-Thequasiparticleoperatorsψky↑,ψky↓canbeexpressedinactionterm(Leggett,1975),sinceitdoesnotaffectanytermsoftheeigenstatesuky(x),vky(x)oftheBdGHamilto-essentialtopologicalproperties.Performingaspinrotation,nianas∆αjcanbediagonalizedto∆αj=∆δαj,inwhichcasetheZHamiltonian(92)canbeexpressedas:ψ=d2xu(x)c(x)+v(x)c†(x),ky↑ky↑ky↑ǫ0∆p−∆pp+zZZ2∗∗†12†0ǫp−∆pz−∆p−ψky↓=dxu−ky(x)c↓(x)+v−ky(x)c↓(x),H=2dxΨ∗∗(93)Ψ.∆p−−∆pz−ǫp0−∆∗p−∆∗p0−ǫfromwhichtheTRtransformationofthequasiparti-z+pcleoperatorscanbedeterminedtobeTψT−1=ComparedwiththemodelHamiltonianEq.(31)forthesim-ky↑ψ,TψT−1=−ψ.Inotherwords,plest3Dtopologicalinsulators(Zhangetal.,2009),wesee−ky↓ky↓−ky↑ 45thattheHamiltonian(93)hasthesameformasthatforBi2Se3(uptoabasistransformation),butwithcomplexfermionsreplacedbyMajoranafermions.Thekineticenergytermp2/2m−µcorrespondstothemomentumdependentmasstermM(p)=M−Bp2−Bp2ofthetopologicalinsulator.1z2kTheweakpairingphaseµ>0correspondstothenontriv-ialtopologicalinsulatorphase,andthestrongpairingphaseµ<0correspondstothetrivialinsulator.Fromthisanalogy,weseethatthesuperconductorHamiltonianintheweakpair-ingphasedescribesatopologicalsuperconductorwithgaplesssurfacestatesprotectedbyTRsymmetry.Differentfromthetopologicalinsulator,thesurfacestatesofthetopologicalsu-perconductorareMajoranafermionsdescribedby1XH=vψT(kσ−kσ)ψ,(94)FIG.41SettingfordetectingtheMajoranasurfacestatesofthe3He-surfF−kxyyxk2kBphase,whichconsistofasingleMajoranacone.Whenelectrons3areinjectedintoHe-B,theyexistas“bubbles”.Iftheinjectedelec-†Ttronsarespin-polarized,thespinwillrelaxbyinteractionwiththewiththeMajoranaconditionψ−k=σxψk.WeseethatsurfaceMajoranamodes,andthisrelaxationisstronglyanisotropic.thisHamiltonianforthesurfaceMajoranafermionsofatopo-FromChungandZhang,2009.logicalsuperconductortakesthesameformasthesurfaceDiracHamiltonianofatopologicalinsulatorinthisspecialbasis.However,becauseofthegenericparticle-holesymme-spaceastryoftheBdGHamiltonianforsuperconductors,thepossi-bleparticle-holesymmetrybreakingtermsforsurfaceDiracX1†††Tfermionssuchasafinitechemicalpotentialisabsentforsur-H=ψkhkψk+ψk∆kψ−k+h.c.,2faceMajoranafermions.Becauseoftheparticle-holesym-kmetryandTRsymmetry,thespinliesstrictlyintheplaneP†perpendiculartothesurfacenormal,andtheintegerwind-InadifferentbasiswehaveH=kΨkHkΨkwithingnumberofthespinaroundthemomentumisnowawell-!†definedquantity.ThisintegerwindingnumbergivesaZclas-√1ψk−iTψ−ksificationofthe3Dtopologicalsuperconductor(Kitaev,2009;Ψk=†,2ψk+iTψ−kSchnyderetal.,2008).Thesurfacestateremainsgaplessun-!†deranysmallTRinvariantperturbation,sincetheonlyavail-10hk+iT∆kPTyHk=2†.(95)ablemasstermmkψ−kσψkisTRodd.TheMajoranahk−iT∆k0surfacestateisspin-polarized,andcanthusbedetectedbyitsspecialcontributiontothespinrelaxationofanelectrononIngeneral,ψkisavectorwithNcomponents,andhkand∆kthesurfaceof3He-B,similartothemeasurementofelectronareN×Nmatrices.ThematrixTistheTRmatrixsatisfyingspincorrelationinasolidstatesystembynuclearmagneticT†hT=hT,T2=−IandT†T=I,withItheiden-k−kresonance(ChungandZhang,2009).titymatrix.WehavechosenaspecialbasisinwhichtheBdGHamiltonianHkhasaspecialoff-diagonalform.ItshouldbenotedthatsuchachoiceisonlypossiblewhenthesystemhasbothTRsymmetryandparticle-holesymmetry.ThesetwoB.Topologicalinvariants†symmetriesalsorequireT∆tobeHermitian,whichmakesk†Fromthediscussionabove,weseethatthemodelHamil-thematrixhk+iT∆kgenericallynon-Hermitian.Thematrix†tonianforthetopologicalsuperconductoristhesameasthathk+iT∆kcanbedecomposedbyasingularvaluedecompo-††forthetopologicalinsulator,butwiththeadditionalparticle-sitionashk+iT∆k=UkDkVkwithUk,Vkunitarymatri-holesymmetry.ThesimultaneouspresenceofbothTRandcesandDkadiagonalmatrixwithnonnegativeelements.Oneparticle-holesymmetrygivesadifferentclassificationforthecanseethatthediagonalelementsofDkareactuallythepos-2Dand3Dtopologicalsuperconductors,inthatthe3DTRin-itiveeigenvaluesofHk.Forafullygappedsuperconductor,varianttopologicalsuperconductorsareclassifiedbyintegerDkispositivedefinite,andwecanadiabaticallydeformitto(Z)classes,andthe2DTRinvarianttopologicalsupercon-theidentitymatrixIwithoutclosingthesuperconductinggap.†ductorsareclassifiedbytheZ2classes.Todefineaninteger-Duringthisdeformation,thematrixhk+iT∆kisdeformed†valuedtopologicalinvariant(Schnyderetal.,2008),westarttoaunitarymatrixQk=UkVk∈U(N).Theinteger-valuedfromagenericmean-fieldBdGHamiltonianfora3DTRin-topologicalinvariantcharacterizingtopologicalsuperconduc-variantsuperconductor,whichcanbewritteninmomentumtorsisdefinedasthewindingnumberofQk(Schnyderetal., 46C=±1[Fig.42(a),(b)].Ifwechoose∆=i∆σy,whichkkz±k0(a)zN=1(b)N=0hasthesameδsforbothFermisurfaces,weobtainNW=0[Fig.42(b)].Ifweinsteadchoose∆=i∆σyσ·k,weob-k02121tainN=1[Fig.42(a)].Inthelattercase,ifwetaketheWα→0limit,wearriveattheresultN=1forthe3He-BWphase,whichindicatesthat3He-Bistopologicallynontriv--ky+kykx+kx+ial(Volovik,2003).For2DTRinvariantsuperconductors,aprocedureofdi-(c)(d)EmensionalreductionleadstothefollowingsimpleFermisur-facetopologicalinvariant:143mY1ms2N2D=(sgn(δs)).(99)sm243m3Thecriterion(99)isquitesimple:a2DTRinvariantsuper-k0pconductoristopologicallynontrivial(trivial)ifthereisanodd(even)numberofFermisurfaces,eachofwhichenclosesoneTRinvariantpointintheBrillouinzoneandhasnegativepair-FIG.42(a,b)SuperconductingpairingontwoFermisurfacesofaing.Asanexample,seeFig.42(c),whereFermisurfaces23Dsuperconductor.(c)Anexampleof2DTRinvarianttopologi-and3havenegativepairing.Fermisurfaces3and4enclosecalsuperconductor.(d)1DTRinvarianttopologicalsuperconductor.anevennumberofTRinvariantmomenta,whichdonotaffectAdaptedfromQietal.,2010.theZ2topologicalinvariant.ThereisonlyoneFermisurface,surface2,whichenclosesonoddnumberofTRinvariantmo-2008):mentaandhasnegativepairing.Asaresult,theZ2topologicalinvariantis(−1)1=−1.ZhiN=1d3kǫijkTrQ†∂QQ†∂QQ†∂Q(96).For1DTRinvariantsuperconductors,afurtherdimensionalW24π2kikkjkkkkreductioncanbecarriedouttogiveWenotethatthetopologicalinvariant(96)isexpressedasYN1D=(sgn(δs))(100)anintegralovertheentireBrillouinzone,similartoitscoun-sterpartfortopologicalinsulators.However,thereisakeydifference.WhereastheinsulatinggapiswelldefinedoverwheresissummedoveralltheFermipointsbetween0andtheentireBrillouinzone,thesuperconductingpairinggapinπ.Ingeometricalterms,a1DTRinvariantsuperconductoristheBdGequationisonlywelldefinedclosetotheFermisur-nontrivial(trivial)ifthereisanoddnumberofFermipointsface.Indeed,superconductivityarisesfromaFermisurfacebetween0andπwithnegativepairing.Weillustratethisfor-instability,atleastintheBCSlimit.Therefore,onewouldmulainFig.42(d),wherethesignofpairingonthered(blue)liketodefinetopologicalinvariantsforatopologicalsuper-Fermipointis−1(+1),sothatthenumberofFermipointsconductorstrictlyintermsofFermisurfacequantities.Thewithnegativepairingis1ifthechemicalpotentialµ=µ1ordesiredtopologicalinvariantcanbeobtainedbyreducingtheµ=µ2,and0ifµ=µ3.ThesuperconductingstateswithwindingnumberinEq.(96)toaintegralovertheFermisur-µ=µ1andµ=µ2canbeadiabaticallydeformedtoeachface(Qietal.,2010):otherwithoutclosingthegap.However,thesuperconductor1Xwithµ=µ3canonlybeobtainedfromthatwithµ2throughNW=sgn(δs)C1s,(97)atopologicalphasetransition,wherethepairingorderparam-2seterchangessignononeoftheFermipoints.Itiseasytoseefromthisexamplethattherearetwoclassesof1DTRinvari-wheresissummedoveralldisconnectedFermisurfacesandantsuperconductors.sgn(δs)denotesthesignofthepairingamplitudeonthesthFermisurface.C1sisthefirstChernnumberofthesthFermisurface(denotedbyFSs):C.MajoranazeromodesintopologicalsuperconductorsZ1ijC1s=dΩ(∂iasj(k)−∂jasi(k)),(98)1.Majoranazeromodesinp+ipsuperconductors2πFSswithasi=−ihsk|∂/∂ki|skitheadiabaticconnectionde-BesidesthenewTRinvarianttopologicalsuperconductors,finedfortheband|skiwhichcrossestheFermisurface,andtheTRbreakingtopologicalsuperconductorshaveattractedadΩijthesurfaceelement2-formoftheFermisurface.lotofinterestbecauseoftheirrelevancetonon-Abelianstatis-Asanexample,weconsideratwo-bandmodelwithnonin-ticsandtopologicalquantumcomputation.Inap+ipsuper-teractingHamiltonianh=k2/2m−µ+αk·σ,forwhichconductordescribedbyEq.(88),itcanbeshownthatthecorektherearetwoFermisurfaceswithoppositeChernnumberofasuperconductingvortexcontainsalocalizedquasiparticle 47withexactlyzeroenergy(Volovik,1999).Thecorrespond-2.MajoranafermionsinsurfacestatesofthetopologicalingquasiparticleoperatorγisaMajoranafermionobeyinginsulator[γ,H]=0andγ†=γ.Whentwovorticeswindaroundeachother,thetwoMajoranafermionsγ1,γ2intheircoresFuandKane(FuandKane,2008)proposedawaytoreal-transformnontrivially.Becausethephaseofthecharge-2eizetheMajoranazeromodeinasuperconductingvortexcoreorderparameterwindsby2πaroundeachvortex,anelec-bymakinguseofthesurfacestatesof3Dtopologicalinsula-tronacquiresaBerryphaseofπwhenwindingoncearoundtors.ConsideratopologicalinsulatorsuchasBi2Se3,whichavortex.SincetheMajoranafermionoperatorisasuperpo-hasasingleDiracconeonthesurfacewithHamiltonianfromsitionofelectroncreationandannihilationoperators,italsoEq.34:acquiresaπphaseshift,i.e.aminussignwhenwindingXH=ψ†[v(σ×p)·ˆz−µ]ψ,(101)aroundanothervortex.Consequently,whentwovorticesareexchanged,theMajoranaoperatorsγ1,γ2musttransformaspγ1→γ2,γ2→−γ1.Theadditionalminussignmaybeas-Tsociatedwithγ1orγ2,butnotboth,sothatafterafullwind-whereψ=(ψ↑,ψ↓)andwehavetakenintoaccountafi-ingwehaveγ1(2)→−γ1(2).SincetwoMajoranafermionsnitechemicalpotentialµ.Considernowthesuperconduct-γ1andγ2defineonecomplexfermionoperatorγ1+iγ2,ingproximityeffectofaconventionals-wavesuperconduc-thetwovorticesactuallysharetwointernalstateslabeledbytoronthe2Dsurfacestates,whichleadstothepairingterm††iγ1γ2=±1.Whenthereare2Nvorticesinthesystem,theH∆=∆ψ↑ψ↓+h.c.TheBdGHamiltonianisgivenbycorestatesspana2N-dimensionalHilbertspace.Thebraid-H=1PΨ†HΨ,whereΨ†≡†andBdG2ppψψingofvorticesleadstonon-Abelianunitarytransformations!inthisHilbertspace,implyingthatthevorticesinthissystemyv(σ×p)·zˆ−µiσ∆obeynon-Abelianstatistics(Ivanov,2001;ReadandGreen,Hp≡.−iσy∆∗−v(σ×p)·ˆz+µ2000).Becausetheinternalstatesofthevorticesarenotlocal-izedoneachvortexbutsharedinanonlocalfashionbetweenThevortexcoreofsuchasuperconductorhasbeenshowntothevortices,thecouplingoftheinternalstatetotheenviron-haveasingleMajoranazeromode,similartoap+ipsuper-mentisexponentiallysmall.Asaresult,thesuperpositionconductor(FuandKane,2008;JackiwandRossi,1981).Toofdifferentinternalstatesisimmunetodecoherence,whichunderstandthisphenomenon,onecanconsiderthecaseoffi-isidealforthepurposeofquantumcomputation.Quantumniteµ,andintroduceaTRbreakingmasstermmσzinthecomputationwithtopologicallyprotectedq-bitsisgenerallysurfacestateHamiltonian(101).AsdiscussedinSec.III.B,knownastopologicalquantumcomputation,andiscurrentlythisopensagapofmagnitude|m|onthesurface.Consider-anactivefieldofresearch(Nayaketal.,2008).ingthecaseµ>m>0,µ−m≪m,theFermilevelinthenormalstateliesnearthebottomoftheparabolicdispersion,andwecanconsidera“nonrelativisticapproximation”totheSeveralexperimentalcandidatesforp-wavesuperconduc-massiveDiracHamiltonian,tivityhavebeenproposed,amongwhichSr2RuO4,whichXH=ψ†[v(σ×p)·ˆz+mσz−µ]ψisconsideredasthemostpromisingcandidatefor2Dchi-ralsuperconductivity(MackenzieandMaeno,2003).How-pZever,manypropertiesofthissystemremainunclear,suchasp22†whetherthissuperconductingphaseisgappedandwhether≃dxψ++m−µψ+,(102)2mtherearegaplessedgestates.whereψ+isthepositiveenergybranchofthesurfacestates.qInmomentumspace,ψ+p=qupψ↑+vpψ↓withup=1+√mandv=p+1−√m.Consider-Fortunately,thereisanalternateroutetowardstopological22p2+m2p|p|22p2+m2superconductivitywithoutp-wavepairing.In1981,JackiwingtheprojectionofthepairingtermH∆ontotheψ+band,andRossi(JackiwandRossi,1981)showedthataddingaMa-weobtainjoranamasstermtoasingleflavorofmasslessDiracfermionsX††in(2+1)DwouldleadtoaMajoranazeromodeinthevortexH∆≃ψ+,pψ+,−p∆upvp+h.c.core.SuchaMajoranamasstermcanbenaturallyinterpretedpasthepairingfieldduetotheproximitycouplingtoaconven-X∆p+††tionals-wavesuperconductor.Therearenowthreedifferent≃ψ+,pψ+,−p+h.c.(103)2mpproposalstorealizethisroutetowardstopologicalsupercon-ductivity:thesuperconductingproximityeffectonthe2Dsur-Weseethatinthislimit,thesurfaceHamiltonianisthesamefacestateofthe3Dtopologicalinsulator(FuandKane,2008),asthatofaspinlessp+ipsuperconductor[Eq.(88)].Whenonthe2DTRbreakingtopologicalinsulator(Qietal.,2010),themassmisturnedonfromzerotoafinitevalue,itcanbeandonsemiconductorswithstrongRashbaSOC(Sauetal.,shownthataslongasm<µ,thesuperconductinggapnear2010).Weshallreviewthesethreeproposalsinthefollowing.theFermisurfaceremainsfinite,sothattheMajoranazero 48modeweobtainedinthelimit0<µ−m≪mmustre-mainatzeroenergyfortheoriginalm=0system.OncewehaveshowntheexistenceofaMajoranazeromodeatfiniteµ,takingtheµ→0limitforafinite∆alsoleavesthesuper-conductinggapopen,sothattheMajoranazeromodeisstillpresentatµ=0.Fromtheanalogywiththep+ipsuperconductorshownabove,wealsoseethatthenon-AbelianstatisticsofvorticeswithMajoranazeromodesapplytothisnewsystemaswell.Akeydifferencebetweenthissystemandachiralp+ipsuper-conductoristhatthelatternecessarilybreaksTRsymmetrywhiletheformercanbeTRinvariant.Onlyaconventionals-wavesuperconductorisrequiredtogeneratetheMajoranazeromodesinthisproposalandintheotherproposalsdis-cussedinthefollowingsubsection.Thisisanimportantad-vantagecomparedtopreviousproposalsrequiringanuncon-ventionalp+ippairingmechanism.Thereisalsoalowerdimensionalanalogofthisnontrivialsurfacestatesuperconductivity.Whentheedgestatesof2DQSHinsulatorareinproximitywithans-wavesuperconduc-torandaferromagneticinsulator,oneMajoranafermionap-pearsateachdomainwallbetweenferromagneticregionandsuperconductingregion((FuandKane,2009)).TheMajo-ranafermioninthissystemcanonlymovealongthe1DQSHFIG.43(a)PhasediagramoftheQAH-superconductorhybridsys-edge,sothatnon-Abelianstatisticsisnotwell-defined.Be-temforµ=0.misthemassparameter,∆isthemagnitudeofthecauseanelectroncannotbebackscatteredontheQSHedge,superconductinggap,andNistheChernnumberofthesupercon-ductor,whichisequaltothenumberofchiralMajoranaedgemodes.thescatteringoftheedgeelectronbyasuperconductingre-(b)Phasediagramforfiniteµ,shownonlyfor∆≥0.TheQAH,gioninducedbyproximityeffectisalwaysperfectAndreevnormalinsulator(NI)andmetallic(Metal)phasesarewell-definedreflection(Adrogueretal.,2010;GuigouandCayssol,2010;onlyfor∆=0.FromQietal.,2010.Satoetal.,2010).perpendiculardirection,becausethemagneticfieldmayde-3.MajoranafermionsinsemiconductorswithRashbastroysuperconductivity.TwowaystorealizeaTRbreakingspin-orbitcouplingmasstermhavebeenproposed:byapplyinganin-planemag-neticfieldandmakinguseoftheDresselhausSOC(Alicea,Fromtheaboveanalysis,weseethatconventionals-pairing2010),orbyexchangecouplingtoaferromagneticinsulatinginthesurfaceHamiltonian(101)inducestopologicallynon-layer(Sauetal.,2010).Thelatterproposalrequiresahet-trivialsuperconductivitywithMajoranafermions.Thereisaerostructureconsistingofasuperconductor,a2Delectrongas2DsystemwhichisdescribedbyaHamiltonianverysimilarwithRashbaSOC,andamagneticinsulator.toEq.(101),i.e.a2DelectrongaswithRashbaSOC.TheThismechanismcanalsobegeneralizedtothe1Dsemicon-R2HamiltonianisH=d2xψ†p+α(σ×p)·zˆ−µψ,2mductorwireswithRashbaSOCcouplinginproximitywithawhichdiffersfromthesurfacestateHamiltonianonlybysuperconductor(Oregetal.,2010;Wimmeretal.,2010).De-thespin-independenttermp2/2m.Consequently,whenspitethe1Dnatureofthewires,non-Abelianstatisticsisstillconventionals-wavepairingisintroduced,eachofthetwopossiblebymakinguseofwirenetworks(Aliceaetal.,2010).spin-splitFermisurfacesformsanontrivialsuperconductor.However,theMajoranafermionsfromthesetwoFermisur-facesannihilateeachothersothatthes-wavesuperconduc-4.MajoranafermionsinquantumHallandquantumtorintheRashbasystemistrivial.Itwaspointedoutre-anomalousHallinsulatorscently(Sauetal.,2010)thatanontrivialsuperconductingphasecanbeobtainedbyintroducingaTRbreakingtermMorerecently,anewapproachtorealizeatopologicalMσzintotheHamiltonian,whichsplitsthedegeneracynearsuperconductorphasehasbeenproposed(Qietal.,2010),k=0.Ifthechemicalpotentialistunedto|µ|<|M|,thewhichisbasedontheproximityeffecttoa2DQHorQAHinnerFermisurfacedisappears.Therefore,superconductivityinsulator.IntegerQHstatesareclassifiedbyanintegerNisonlyinducedbypairingontheouterFermisurface,andbe-correspondingtothefirstChernnumberinmomentumspacecomestopologicallynontrivial.Physically,onecannotinduceandequaltotheHallconductanceinunitsofe2/h.Con-aTRbreakingmasstermbyapplyingamagneticfieldinthesideraQHinsulatorwithHallconductanceNe2/hinclose 49proximitytoasuperconductor.Evenifthepairingstrengthinducingbythesuperconductingproximityeffectisinfinites-imallysmall,theresultingstateistopologicallyequivalenttoachiraltopologicalsuperconductorwithZtopologicalquan-tumnumberN=2N.AnintuitivewaytounderstandsucharelationbetweenQHandtopologicalsuperconductingphasesisthroughtheevolutionoftheedgestates.TheedgestateofaQHstatewithChernnumberN=1isdescribedPbytheeffective1DHamiltonianH=vpη†η,edgepyypypywhereη†,ηarecreation/annihilationoperatorsforacom-pypyplexspinlessfermion.Wecandecomposeηpyintoitsreal√andimaginaryparts,η=1/2(γ+iγ)andη†=√pypy1py2py1/2(γ−py1−iγ−py2),whereγpyaareMajoranafermionno†operatorssatisfyingγpya=γ−pyaandγ−pya,γp′b=yδabδp′.TheedgeHamiltonianbecomesypyX
Hedge=pyγ−py1γpy1+γ−py2γpy2,(104)py≥0uptoatrivialshiftoftheenergy.IncomparisonwiththeedgeFIG.443Dtopologicalinsulatorinproximitytoferromagnetstheoryofthechiraltopologicalsuperconductingstate,theQHwithoppositepolarization(M↑andM↓)andtoasuperconductoredgestatecanbeconsideredastwoidenticalcopiesofchiral(S).ThetoppanelshowsasinglechiralMajoranamodealongtheMajoranafermions,sothattheQHphasewithChernnumberedgebetweensuperconductorandferromagnet.Thismodeiselec-tricallyneutral,andthereforecannotbedetectedelectrically.TheN=1canbeconsideredasachiraltopologicalsupercon-Mach-ZehnderinterferometerinthebottompanelconvertsachargedductingstatewithChernnumberN=2,evenforinfinitesi-currentalongthedomainwallintoaneutralcurrentalongthesu-malpairingamplitudes.perconductor(andviceversa).ThisallowselectricaldetectionofAnimportantconsequenceofsucharelationbetweenQHtheparityofthenumberofenclosedvortices/fluxquanta.FromandtopologicalsuperconductingphasesisthattheQHplateauAkhmerovetal.,2009.transitionfromN=1toN=0willgenericallysplitintotwotransitionswhensuperconductingpairingisintroduced.Betweenthetwotransitions,therewillbeanewtopologi-5.DetectionofMajoranafermionscalsuperconductingphasewithoddwindingnumberN=1[Fig.43].Comparedtootherapproaches,theemergenceofThenextobviousquestionishowtodetecttheMa-thetopologicalsuperconductingphaseataQHplateautran-joranafermionifsuchaproposalisexperimentallyre-sitionisdeterminedtopologically,sothatthisapproachdoesalized.Thereexisttwosimilartheoreticalproposalsofnotdependonanyfinetuningordetailsofthetheory.electricaltransportmeasurementstodetecttheseMajoranaAnaturalconcernraisedbythisapproachisthatthestrongfermions(Akhmerovetal.,2009;FuandKane,2009).Con-magneticfieldusuallyrequiredforQHstatescansuppresssu-siderthegeometryshowninFig.44.Thisdeviceisacom-perconductivity.ThesolutiontothisproblemcanbefoundbinationoftheinhomogeneousstructuresonthesurfaceofinaspecialtypeofQHstate—theQAHstate,whichatopologicalinsulatordiscussedintheprevioussubsections.isaTRbreakinggappedstatewithnonzeroHallconduc-Theinputandoutputofthecircuitconsistofachiralfermiontanceintheabsenceofanexternalorbitalmagneticfieldcomingfromadomainwallbetweentwoferromagnets.This(Sec.II.E).Thereexistnowtworealisticproposalsforreal-chiralfermionisincidentonasuperconductingregionwhereizingtheQAHstateexperimentally,bothofwhichmakeuseitsplitsintotwochiralMajoranafermions.ThechiralMa-oftheTRinvarianttopologicalinsulatormaterialsMn-dopedjoranafermionsthenrecombineintoanoutgoingelectronorHgTeQWs(Liuetal.,2008),andCr-orFe-dopedBi2Se3thinholeaftertravelingaroundthesuperconductingisland.Morefilms(Yuetal.,2010).Thelattermaterialisproposedtobeexplicitly,anelectronincidentfromthesourcecanbetrans-ferromagnetic,andcanthusexhibitaquantizedHallconduc-mittedtothedrainasanelectron,orconvertedtoaholebytanceatzeromagneticfield.TheformermaterialisknowntoanAndreevprocessinwhichcharge2eisabsorbedintothebeparamagneticforlowMnconcentrations,butonlyasmallsuperconductingcondensate.ToillustratetheideawediscussmagneticfieldisneededtopolarizetheMnspinsanddrivethebehaviorforaE=0quasiparticle(FuandKane,2009).thesystemintoaQAHphase.Thisrequirementisnotsopro-Achiralfermionincidentatpointameetsthesuperconduc-hibitive,becauseanonzeromagneticfieldisalreadyneces-torandevolvesfromanelectronc†intoafermionψbuiltasarytogeneratesuperconductingvorticesandtheassociatedfromtheMajoranaoperatorsγ1andγ2.ThearbitrarinessinMajoranazeromodes.thesignofγ1,2allowsustochooseψ=γ1+iγ2.After 50thequasiparticlewindsaroundthesuperconductingregion,ψordereffects,realisticpredictionsfortopologicalMottinsu-recombinesintoacomplexfermionatpointd.Thisfermionlatormaterials,adeeperunderstandingoffractionaltopolog-†mustbeeithercdorcd,sinceasuperpositionofthetwoisnotaicalinsulatorsandrealisticpredictionsformaterialsrealiza-fermionoperatorandisthusforbidden.Todeterminethecor-tionsofsuchstates,theeffectivefieldtheorydescriptionofrectoperatorwecanuseadiabaticcontinuity.Whenthesizethetopologicalsuperconductingstate,andrealisticmaterialsofthesuperconductorshrinkscontinuouslytozero,pointsapredictionsfortopologicalsuperconductors.Ontheexperi-anddcontinuouslytendtoeachother.Adiabaticcontinuitymentalside,themostimportanttaskistogrowmaterialswithimpliesthatanincidentE=0electronistransmittedasansufficientpuritysothatthebulkinsulatingbehaviorcanbeelectron,c†→c†.However,iftheringenclosesaquantizedreached,andtotunetheFermilevelclosetotheDiracpointadfluxΦ=nhc/2e,thisadiabaticargumentmustbereconsid-ofthesurfacestate.Hybridstructuresbetweentopologicalin-ered.Whennisanoddinteger,thetwoMajoranafermionssulatorsandmagneticandsuperconductingstateswillbein-acquireanadditionalrelativephaseofπ,sinceeachfluxquan-tensivelyinvestigated,withafocusondetectingexoticemer-tumhc/2eisaπfluxforanelectron,andthusπforaMajo-gentparticlessuchastheimagemagneticmonopole,theaxionranafermion.Uptoanoverallsign,onecantakeγ1→−γ1andtheMajoranafermion.Thetheoreticalpredictionoftheandγ2→γ2.Thus,whentheringenclosesanoddnumberofQAHstateissufficientlyrealisticanditsexperimentaldiscov-fluxquanta,c†→c,andanincidentE=0electroniscon-eryappearstobeimminent.Thetopologicalquantizationofadvertedtoahole.Thegeneralconsequencesofthiswerecal-theTMEeffectin3Dandthespin-chargeseparationeffectculatedindetail(Akhmerovetal.,2009;FuandKane,2009),in2Dcouldexperimentallydeterminethetopologicalorderanditwasshownthattheoutputcurrent(througharmdintheparameterofthisnovelstateofmatter.lowerpanelofFig.44)changessignwhenthenumberoffluxDuetospacelimitations,wedidnotdiscussindetailthequantaintheringjumpsbetweenoddandeven.Thisuniquepotentialforapplicationsoftopologicalinsulatorsandsuper-behaviorofthecurrentprovidesawaytoelectricallydetectconductors.ItwouldbeinterestingtoexplorethepossibilityMajoranafermions.ofelectronicdeviceswithlowpowerconsumptionbasedonBesidesthesetwoproposalsreviewedabove,severalotherthedissipationlessedgechannelsoftheQSHstate,spintron-theoreticalproposalshavealsobeenmaderecentlytoob-icsdevicesbasedontheuniquecurrent-spinrelationshipinservetheMajoranafermionstate,whichmakeuseofthetopologicalsurfacestates,infrareddetectors,andthermo-theCoulombchargingenergy(Fu,2010)orafluxqubitelectricapplications.Topologicalquantumcomputersbased(Hassleretal.,2010).Moreindirectly,MajoranafermionsonMajoranafermionsremainagreatinspirationinthefield.canalsobedetectedthroughtheircontributiontoJosephsonTopologicalinsulatorsandsuperconductorsofferaplatformcoupling(FuandKane,2008,2009;LinderandSudbo,2010;totestmanynovelideasinparticlephysics—a“babyuni-Linderetal.,2010;Lutchynetal.,2010;Tanakaetal.,2009).verse”wherethemysteriousθvacuumisrealized,whereex-ForatopologicalsuperconductorringwithMajoranafermionsoticparticlesroamfreelyandwherecompactifiedextradi-atbothends,theperiodofJosephsoncurrentisdoubled,in-mensioncanbetestedexperimentally.Intheintroductiontodependentfromthephysicalrealization(FuandKane,2009;thisarticlewedrewananalogybetweenthesearchfornewKitaev,2001;Lutchynetal.,2010).statesofmatterandthediscoveryofelementaryparticles.Uptonow,themostimportantstatesofquantummatterwerefirstdiscoveredempiricallyandoftenserendipitously.OntheotherVI.OUTLOOKhand,theEinstein-Diracapproachhasbeenmostsuccessfulinsearchingforthefundamentallawsofnature:purelogicalThesubjectoftopologicalinsulatorsandtopologicalsuper-reasoningandbeautifulmathematicalequationsguidedandconductorsisnowoneofthemostactivefieldsofresearchinpredictedsubsequentexperimentaldiscoveries.Thesuccesscondensedmatterphysics,developingatarapidpace.The-oftheoreticalpredictionsinthefieldoftopologicalinsulatorsoristshavesystematicallyclassifiedtopologicalstatesinallshowsthatthispowerfulapproachworksequallywellincon-dimensions.Ref.(Qietal.,2008)initiatedtheclassifica-densedmatterphysics,hopefullyinspiringmanymoreexam-tionprogramofalltopologicalinsulatorsaccordingtodiscreteplestocome.particle-holesymmetryandtheTRsymmetry,andnoticedaperiodicstructurewithperiodeight,whichisknowninmath-ematicsastheBottperiodicity.MoreextendedandsystematicACKNOWLEDGMENTSclassificationofalltopologicalinsulatorandsuperconductorstatesareobtainedaccordingtoTR,particle-holeandbipartiteWearedeeplygratefultoTaylorL.Hughes,Chao-Xingsymmetries(Kitaev,2009;Qietal.,2008;Ryuetal.,2010;Liu,JosephMaciejkoandZhongWangfortheirinvaluableSchnyderetal.,2008;Stoneetal.,2010).Suchclassificationinputswhichmadethecurrentmanuscriptpossible.Weschemegivesa“periodictable”oftopologicalstates,whichwouldliketothankAndreiBernevig,HartmutBuhmann,mayplayasimilarroleasthefamiliarperiodtableofele-YulinChen,SukBumChung,YiCui,XiDai,DennisDrew,ments.Forfutureprogressonthetheoreticalside,themostZhongFang,AharonKapitulnik,AndreasKarch,Laurensimportantoutstandingproblemsincludeinteractionanddis-Molenkamp,NaotoNagaosa,S.Raghu,Zhi-XunShen,Cenke 51Xu,QikunXue,HaijunZhangfortheirclosecollaborationCastroNetoA.H.,F.Guinea,N.M.R.Peres,K.S.Novoselov,andandfortheirimportantcontributionsreviewedinthispaper.A.K.Geim(2009),Rev.Mod.Phys.,81,109.WebenefittedgreatlyfromthediscussionswithcolleaguesChaJ.J.,J.R.Williams,D.Kong,S.Meister,H.Peng,A.J.Best-wick,P.Gallagher,D.Goldhaber-Gordon,andY.Cui(2010),LeonBalents,MacBeasley,CarloBeenakker,MarcelFranz,NanoLett.,10,1076.LiangFu,DavidGoldhaber-Gordon,ZahidHasan,CharlieChadovS.,X.L.Qi,J.K¨ubler,G.H.Fecher,C.Felser,andS.C.Kane,AlexeiKitaev,SteveKivelson,AndreasLudwig,JoelZhang(2010),NatureMater.,9,541.Moore,PhuanOng,RahulRoy,ShinseiRyu,AliYazdaniCheckelskyJ.G.,Y.S.Hor,R.J.Cava,andN.P.Ong(2010),e-printandJanZaanen.ThisworkissupportedbytheDepartmentarXiv:1003.3883.ofEnergy,OfficeofBasicEnergySciences,DivisionofMa-CheckelskyJ.G.,Y.S.Hor,M.-H.Liu,D.-X.Qu,R.J.Cava,andterialsSciencesandEngineering,undercontractDE-AC02-N.P.Ong(2009),Phys.Rev.Lett.,103,246601.ChenJ.,H.J.Qin,F.Yang,J.Liu,T.Guan,F.M.Qu,G.H.Zhang,76SF00515,theNSFunderthegrantnumberDMR-0904264,J.R.Shi,X.C.Xie,C.L.Yang,K.H.Wu,Y.Q.Li,andL.LutheAROunderthegrantnumberW911NF-09-1-0508andthe(2010a),e-printarXiv:1003.1534.KeckFoundation.ChenL.,andS.Wan(2010),e-printarXiv:1005.4143.ChenY.L.,J.G.Analytis,J.H.Chu,Z.K.Liu,S.-K.Mo,X.L.Qi,H.J.Zhang,D.H.Lu,X.Dai,Z.Fang,S.C.Zhang,I.R.Fisher,Z.Hussain,andZ.-X.Shen(2009),Science,325,178.REFERENCESChenY.L.,J.-H.Chu,J.G.Analytis,Z.K.Liu,K.Igarashi,H.-H.Kuo,X.L.Qi,S.K.Mo,R.G.Moore,D.H.Lu,M.Hashimoto,AbrahamsE.,P.W.Anderson,D.C.Licciardello,andT.V.Ramakr-T.Sasagawa,S.C.Zhang,I.R.Fisher,Z.Hussain,andZ.X.Shenishnan(1979),Phys.Rev.Lett.,42,673.(2010b),unpublished.AdroguerP.,C.Grenier,D.Carpentier,J.Cayssol,P.Degiovanni,ChenY.L.,Z.Liu,J.G.Analytis,J.H.Chu,H.Zhang,S.K.Mo,andE.Orignac(2010),e-printarXiv:1006.0215.R.G.Moore,D.Lu,I.R.Fisher,S.C.Zhang,Z.Hussain,andAkhmerovA.R.,C.W.Groth,J.Tworzydło,andC.W.J.BeenakkerZ.X.Shen(2010c),e-printarXiv:1006.3843.(2009a),Phys.Rev.B,80,195320.ChengP.,C.Song,T.Zhang,Y.Zhang,Y.Wang,J.Jia,J.Wang,AkhmerovA.R.,J.Nilsson,andC.W.J.Beenakker(2009b),Phys.Y.Wang,B.Zhu,X.Chen,X.Ma,K.He,L.Wang,X.Dai,Rev.Lett.,102,216404.Z.Fang,X.C.Xie,X.Qi,C.Liu,S.Zhang,andQ.Xue(2010),AliceaJ.(2010),Phys.Rev.B,81,125318.e-printarXiv:1001.3220.AliceaJ.,Y.Oreg,G.Refael,F.vonOppen,andM.P.A.FisherChuR.L.,J.Li,J.K.Jain,andS.Q.Shen(2009),Phys.Rev.B,80,(2010),e-printarXiv:1006.4395.081102(R).AlpichshevZ.,J.G.Analytis,J.H.Chu,I.R.Fisher,Y.L.Chen,ChungS.B.,andS.C.Zhang(2009),Phys.Rev.Lett.,103,235301.Z.X.Shen,A.Fang,andA.Kapitulnik(2010a),Phys.Rev.Lett.,DaiX.,T.L.Hughes,X.-L.Qi,Z.Fang,andS.-C.Zhang(2008),104,016401.Phys.Rev.B,77,125319.AlpichshevZ.,J.G.Analytis,J.H.Chu,I.R.Fisher,andA.Kapit-DaumerV.,I.Golombek,M.Gbordzoe,E.G.Novik,V.Hock,C.R.ulnik(2010b),e-printarXiv:1003.2233.Becker,H.Buhmann,andL.W.Molenkamp(2003),Appl.Phys.AnalytisJ.G.,J.H.Chu,Y.L.Chen,F.Corredor,R.D.McDonald,Lett.,83,1376.Z.X.Shen,andI.R.Fisher(2010a),Phys.Rev.B,81,205407.DayC.(2008),Phys.Today,61,19.AnalytisJ.G.,R.D.McDonald,S.C.Riggs,J.H.Chu,G.S.Boe-DubrovinB.A.,A.T.Fomenko,andS.P.Novikov(1985),Modernbinger,andI.R.Fisher(2010b),e-printarXiv:1003.1713.Geometry—MethodsandApplications,Part2:TheGeometryandAndersonP.W.(1997),BasicNotionsofCondensedMatterPhysicsTopologyofManifolds(Springer).(WestviewPress).DzeroM.,K.Sun,V.Galitski,andP.Coleman(2010),Phys.Rev.AsanoY.,Y.Tanaka,andN.Nagaosa(2010),e-printLett.,104,106408.arXiv:1004.4092.EssinA.M.,andJ.E.Moore(2007),Phys.Rev.B,76,165307.Ayala-ValenzuelaO.E.,J.G.Analytis,J.H.Chu,M.M.Al-EssinA.M.,J.E.Moore,andD.Vanderbilt(2009),Phys.Rev.Lett.,tarawneh,I.R.Fisher,andR.D.McDonald(2010),e-print102,146805.arXiv:1004.2311.EssinA.M.,A.M.Turner,J.E.Moore,andD.Vanderbilt(2010),BardarsonJ.H.,P.W.Brouwer,andJ.E.Moore(2010),e-printPhys.Rev.B,81,205104.arXiv:1005.3762.EtoK.,Z.Ren,A.A.Taskin,K.Segawa,andY.Ando(2010),Phys.BeenakkerC.W.J.,andH.vanHouten(1991),SolidStatePhys.,Rev.B,81,195309.44,1.EversF.,andA.D.Mirlin(2008),Rev.Mod.Phys.,80,1355.BernevigB.A.,C.H.Chern,J.P.Hu,N.Toumbas,andS.C.ZhangFengX.-Y.,W.-Q.Chen,J.-H.Gao,Q.-H.Wang,andF.-C.Zhang(2002),Ann.Phys.,300,185.(2009),e-printarXiv:0910.3031.BernevigB.A.,T.L.Hughes,andS.C.Zhang(2006),Science,314,FradkinE.,E.Dagotto,andD.Boyanovsky(1986),Phys.Rev.Lett.,1757.57,2967.BernevigB.A.,andS.C.Zhang(2006),Phys.Rev.Lett.,96,FranzM.(2010),NatureMater.,9,536.106802.FreedmanM.,M.B.Hastings,C.Nayak,X.-L.Qi,K.Walker,andBiswasR.R.,andA.V.Balatsky(2010),e-printarXiv:1005.4780.Z.Wang(2010),e-printarXiv:1005.0583.BurkovA.A.,andD.G.Hawthorn(2010),e-printarXiv:1005.1654.FuL.(2009),Phys.Rev.Lett.,103,266801.ButchN.P.,K.Kirshenbaum,P.Syers,A.B.Sushkov,G.S.Jenkins,FuL.(2010),Phys.Rev.Lett.,104(5),056402.H.D.Drew,andJ.Paglione(2010),Phys.Rev.B,81,241301.FuL.,andC.L.Kane(2006),Phys.Rev.B,74,195312.B¨uttikerM.(1986),Phys.Rev.Lett.,57,1761.FuL.,andC.L.Kane(2007),Phys.Rev.B,76,045302.B¨uttikerM.(1988),Phys.Rev.B,38,9375.FuL.,andC.L.Kane(2008),Phys.Rev.Lett.,100,096407.B¨uttikerM.(2009),Science,325,278.FuL.,andC.L.Kane(2009a),Phys.Rev.B,79(16),161408. 52FuL.,andC.L.Kane(2009b),Phys.Rev.Lett.,102,216403.JackiwR.,andC.Rebbi(1976),Phys.Rev.D,13,3398.FuL.,C.L.Kane,andE.J.Mele(2007),Phys.Rev.Lett.,98,JackiwR.,andP.Rossi(1981),Nucl.Phys.B,190,681.106803.JiangH.,L.Wang,Q.-F.Sun,andX.C.Xie(2009),Phys.Rev.B,FukuiT.,andY.Hatsugai(2007),Phys.Rev.B,75,121403(R).80,165316.GaoJ.,W.Chen,X.C.Xie,andF.C.Zhang(2009),Phys.Rev.B,KaneC.L.(2008),NaturePhys.,4,348.80,241302.KaneC.L.,andE.J.Mele(2005a),Phys.Rev.Lett.,95,226801.GarateI.,andM.Franz(2010),Phys.Rev.Lett.,104,146802.KaneC.L.,andE.J.Mele(2005b),Phys.Rev.Lett.,95,146802.GiamarchiT.(2003),QuantumPhysicsinOneDimension(Claren-KaplanD.B.(1992),Phys.Lett.B,288,342.donpress,Oxford).KarchA.(2009),Phys.Rev.Lett.,103,171601.GiamarchiT.,andH.J.Schulz(1988),Phys.Rev.B,37,325.KharitonovM.(2010),e-printarXiv:1004.0194.GoldstoneJ.,andF.Wilczek(1981),Phys.Rev.Lett.,47,986.King-SmithR.D.,andD.Vanderbilt(1993),Phys.Rev.B,47,1651.GoltermanM.F.L.,K.Jansen,andD.B.Kaplan(1993),Phys.Lett.KitaevA.(2009),AIPConf.Proc.,1134,22.B,301,219.KitaevA.Y.(2001),Physics-Uspekhi,44,131.GomesK.K.,W.Ko,W.Mar,Y.Chen,Z.-X.Shen,andH.C.KivelsonS.,andJ.R.Schrieffer(1982),Phys.Rev.B,25,6447.Manoharan(2009),e-printarXiv:0909.0921.vonKlitzingK.,G.Dorda,andM.Pepper(1980),Phys.Rev.Lett.,GrothC.W.,M.Wimmer,A.R.Akhmerov,J.Tworzydło,and45,494.C.W.J.Beenakker(2009),Phys.Rev.Lett.,103,196805.KnezI.,R.R.Du,andG.Sullivan(2010),Phys.Rev.B,81,GuigouM.,andJ.Cayssol(2010),e-printarXiv:1005.1055.201301(R).GuoH.M.(2010),e-printarXiv:1006.5951.KongD.,W.Dang,J.J.Cha,H.Li,S.Meister,H.Peng,Z.Liu,andGuoH.M.,andM.Franz(2009),Phys.Rev.Lett.,103,206805.Y.Cui(2010a),NanoLett.,10,2245.GuoH.-M.,andM.Franz(2010),Phys.Rev.B,81,041102.KongD.,J.C.Randel,H.Peng,J.J.Cha,S.Meister,K.Lai,Y.Chen,GuoH.M.,G.Rosenberg,G.Refael,andM.Franz(2010),e-printZ.-X.Shen,H.C.Manoharan,andY.Cui(2010b),NanoLett.,10,arXiv:1006.2777.329.HaldaneF.D.M.(1988),Phys.Rev.Lett.,61,2015.K¨onigM.(2007),Ph.D.thesis(Universit¨atW¨urzburg).HaldaneF.D.M.,andL.Chen(1983),Phys.Rev.Lett.,53,2591.K¨onigM.,H.Buhmann,L.W.Molenkamp,T.L.Hughes,C.-X.Liu,HanaguriT.,K.Igarashi,M.Kawamura,H.Takagi,andT.SasagawaX.L.Qi,andS.C.Zhang(2008),J.Phys.Soc.Jpn,77,031007.(2010),e-printarXiv:1003.0100.K¨onigM.,S.Wiedmann,C.Br¨une,A.Roth,H.Buhmann,HasanM.Z.,andC.L.Kane(2010),e-printarXiv:1002.3895.L.Molenkamp,X.-L.Qi,andS.-C.Zhang(2007),Science,318,HasslerF.,A.R.Akhmerov,C.-Y.Hou,andC.W.J.Beenakker766.(2010),e-printarXiv:1005.3423.LaForgeA.D.,A.Frenzel,B.C.Pursley,T.Lin,X.Liu,J.Shi,andHikamiS.,A.I.Larkin,andY.Nagaoka(1980),Prog.Theor.Phys.,D.N.Basov(2010),Phys.Rev.B,81,125120.63,707.LandauL.D.,andE.M.Lifshitz(1980),StatisticalPhysics(Perga-HongS.S.,W.Kundhikanjana,J.J.Cha,K.Lai,D.Kong,S.Meister,monPress,Oxford).M.A.Kelly,Z.Shen,andY.Cui(2010),e-printarXiv:1007.1285.LandauL.D.,andE.M.Lifshitz(1984),ElectrodynamicsofCon-ndHorY.S.,J.G.Checkelsky,D.Qu,N.P.Ong,andR.J.CavatinuousMedia,2edition(PergamonPress,Oxford).(2010a),e-printarXiv:1006.0317.LaughlinR.B.(1981),Phys.Rev.B,23,5632.HorY.S.,A.Richardella,P.Roushan,Y.Xia,J.G.Checkelsky,LawK.T.,C.Y.Seng,P.A.Lee,andT.K.Ng(2010),Phys.Rev.B,A.Yazdani,M.Z.Hasan,N.P.Ong,andR.J.Cava(2009),Phys.81,041305.Rev.B,79,195208.LeeD.-H.(2009),Phys.Rev.Lett.,103,196804.HorY.S.,P.Roushan,H.Beidenkopf,J.Seo,D.Qu,J.G.Checkel-LeeW.C.,C.Wu,D.P.Arovas,andS.C.Zhang(2009),Phys.Rev.sky,L.A.Wray,D.Hsieh,Y.Xia,S.Xu,D.Qian,M.Z.Hasan,B,80,245439.N.P.Ong,A.Yazdani,andR.J.Cava(2010b),Phys.Rev.B,81,LeggettA.J.(1975),Rev.Mod.Phys.,47,331.195203.LevinM.,andA.Stern(2009),Phys.Rev.Lett.,103,196803.HorY.S.,A.J.Williams,J.G.Checkelsky,P.Roushan,J.Seo,LiH.D.,Z.Y.Wang,X.Kan,X.Guo,H.T.He,Z.Wang,J.N.Q.Xu,H.W.Zandbergen,A.Yazdani,N.P.Ong,andR.J.CavaWang,T.L.Wong,N.Wang,andM.H.Xie(2010a),e-print(2010c),Phys.Rev.Lett.,104,057001.arXiv:1005.0449.HouC.Y.,E.A.Kim,andC.Chamon(2009),Phys.Rev.Lett.,102,LiJ.,R.-L.Chu,J.K.Jain,andS.-Q.Shen(2009a),Phys.Rev.Lett.,076602.102,136806.HsiehD.,D.Qian,L.Wray,Y.Xia,Y.S.Hor,R.J.Cava,andM.Z.LiR.,J.Wang,X.L.Qi,andS.C.Zhang(2010b),NaturePhys.,6,Hasan(2008),Nature,452,970.284.HsiehD.,Y.Xia,D.Qian,L.Wray,J.H.Dil,F.Meier,J.Oster-LiY.-Y.,G.Wang,X.-G.Zhu,M.-H.Liu,C.Ye,X.Chen,Y.-walder,L.Patthey,J.G.Checkelsky,N.P.Ong,A.V.Fedorov,Y.Wang,K.He,L.-L.Wang,X.-C.Ma,H.-J.Zhang,X.Dai,H.Lin,A.Bansil,D.Grauer,Y.S.Hor,R.J.Cava,andM.Z.Z.Fang,X.-C.Xie,Y.Liu,X.-L.Qi,J.-F.Jia,S.-C.Zhang,andHasan(2009a),Nature,460,1101.Q.-K.Xue(2009b),e-printarXiv:0912.5054.HsiehD.,Y.Xia,D.Qian,L.Wray,J.H.Dil,F.Meier,J.Oster-LinH.,R.S.Markiewicz,L.A.Wray,L.Fu,M.Z.Hasan,andwalder,L.Patthey,A.V.Fedorov,H.Lin,A.Bansil,D.Grauer,A.Bansil(2010a),e-printarXiv:1003.2615.Y.S.Hor,R.J.Cava,andM.Z.Hasan(2009b),Phys.Rev.Lett.,LinH.,L.A.Wray,Y.Xia,S.Xu,S.Jia,R.J.Cava,A.Bansil,and103,146401.M.Z.Hasan(2010b),NatureMater.,9,546.HsiehD.,Y.Xia,L.Wray,D.Qian,A.Pal,J.H.Dil,J.Osterwalder,LinderJ.,andA.Sudbo(2010),e-printarXiv:1004.4629.F.Meier,G.Bihlmayer,C.L.Kane,Y.S.Hor,R.J.Cava,andLinderJ.,Y.Tanaka,T.Yokoyama,A.Sudbo,andN.NagaosaM.Z.Hasan(2009c),Science,323,919.(2010),e-printarXiv:1003.4754.ImuraK.-I.,Y.Kuramoto,andK.Nomura(2009),Phys.Rev.B,80,LinderJ.,T.Yokoyama,andA.Sudbø(2009),Phys.Rev.B,80,085119.205401.IvanovD.A.(2001),Phys.Rev.Lett.,86,268. 53LiuC.-X.,T.L.Hughes,X.-L.Qi,K.Wang,andS.-C.ZhangOstrovskyP.M.,I.V.Gornyi,andA.D.Mirlin(2009),e-print(2008a),Phys.Rev.Lett.,100,236601.arXiv:0910.1338.LiuC.-X.,X.-L.Qi,X.Dai,Z.Fang,andS.-C.Zhang(2008b),Phys.PecceiR.D.,andH.R.Quinn(1977),Phys.Rev.Lett.,38,1440.Rev.Lett.,101,146802.PengH.,K.Lai,D.Kong,S.Meister,Y.Chen,X.L.Qi,S.C.Zhang,LiuC.-X.,X.-L.Qi,H.Zhang,X.Dai,Z.Fang,andS.-C.ZhangZ.X.Shen,andY.Cui(2010),NatureMater.,9,225.(2010a),e-printarXiv:1005.1682.PesinD.A.,andL.Balents(2010),NaturePhys.,6,376.LiuC.-X.,H.Zhang,B.Yan,X.-L.Qi,T.Frauenheim,X.Dai,ProdanE.(2009),Phys.Rev.B,80,125327.Z.Fang,andS.-C.Zhang(2010b),Phys.Rev.B,81,041307.QiX.-L.,T.Hughes,andS.-C.Zhang(2008a),NaturePhys.,4,273.LiuQ.,C.-X.Liu,C.Xu,X.-L.Qi,andS.-C.Zhang(2009),Phys.QiX.-L.,T.Hughes,andS.-C.Zhang(2008b),Phys.Rev.B,78,Rev.Lett.,102,156603.195424.LoringT.A.,andM.B.Hastings(2010),e-printarXiv:1005.4883.QiX.-L.,T.L.Hughes,S.Raghu,andS.-C.Zhang(2009a),Phys.LuH.Z.,W.Y.Shan,W.Yao,Q.Niu,andS.Q.Shen(2010),Phys.Rev.Lett.,102,187001.Rev.B,81,115407.QiX.-L.,T.L.Hughes,andS.-C.Zhang(2010a),e-printLutchynR.M.,J.D.Sau,andS.D.Sarma(2010),e-printarXiv:1003.5448.arXiv:1002.4033.QiX.-L.,T.L.Hughes,andS.-C.Zhang(2010b),Phys.Rev.B,81,MaciejkoJ.,C.-X.Liu,Y.Oreg,X.L.Qi,C.Wu,andS.C.Zhang134508.(2009a),Phys.Rev.Lett.,102,256803.QiX.-L.,R.Li,J.Zang,andS.-C.Zhang(2009b),Science,323,MaciejkoJ.,X.-L.Qi,H.D.Drew,andS.-C.Zhang(2010a),e-print1184.arXiv:1004.2514.QiX.L.,Y.S.Wu,andS.C.Zhang(2006),Phys.Rev.B,74,MaciejkoJ.,X.L.Qi,A.Karch,andS.C.Zhang(2010b),e-print085308.arXiv:1004.3628.QiX.-L.,andS.-C.Zhang(2008),Phys.Rev.Lett.,101,086802.MaciejkoJ.,X.-L.Qi,andS.-C.Zhang(2009b),e-printQiX.L.,andS.C.Zhang(2010),Phys.Today,63,33.arXiv:0907.4515.RachelS.,andK.L.Hur(2010),e-printarXiv:1003.2238.MackenzieA.P.,andY.Maeno(2003),Rev.Mod.Phys.,75,657.RaghuS.,S.B.Chung,X.L.Qi,andS.C.Zhang(2010),Phys.Rev.MalashevichA.,I.Souza,S.Coh,andD.Vanderbilt(2010),NewJ.Lett.,104,116401.Phys.,12,053032.RaghuS.,X.-L.Qi,C.Honerkamp,andS.-C.Zhang(2008),Phys.MeidanD.,andY.Oreg(2005),Phys.Rev.B,72,121312(R).Rev.Lett.,100,156401.MillsD.L.,andE.Burstein(1974),Rep.Prog.Phys.,37,817.RanY.,A.Vishwanath,andD.-H.Lee(2008),Phys.Rev.Lett.,101,MinH.,J.Hill,N.Sinitsyn,B.Sahu,L.Kleinman,andA.MacDon-086801.ald(2006),Phys.Rev.B,74,165310.RanY.,Y.Zhang,andA.Vishwanath(2009),NaturePhys.,5,298.MooreJ.E.(2009),NaturePhys.,5,378.ReadN.,andD.Green(2000),Phys.Rev.B,61,10267.MooreJ.E.,andL.Balents(2007),Phys.Rev.B,75,121306.RedlichA.N.(1984a),Phys.Rev.Lett.,52,18.MurakamiS.(2006),Phys.Rev.Lett.,97,236805.RedlichA.N.(1984b),Phys.Rev.D,29,2366.MurakamiS.,S.Iso,Y.Avishai,M.Onoda,andN.Nagaosa(2007),RestaR.(1994),Rev.Mod.Phys.,66,899.Phys.Rev.B,76,205304.RosenbergG.,andM.Franz(2010),e-printarXiv:1001.3179.MurakamiS.,N.Nagaosa,andS.C.Zhang(2003),Science,301,RosenbergG.,H.M.Guo,andM.Franz(2010),e-print1348.arXiv:1002.3155.MurakamiS.,N.Nagaosa,andS.C.Zhang(2004),Phys.Rev.Lett.,RothA.,C.Br¨une,H.Buhmann,L.W.Molenkamp,J.Maciejko,93,156804.X.-L.Qi,andS.-C.Zhang(2009),Science,325,294.NakaharaM.(1990),Geometry,Topology,andPhysics(A.Hilger).RotheD.G.,R.W.Reinthaler,C.-X.Liu,L.W.Molenkamp,S.C.NayakC.,S.H.Simon,A.Stern,M.Freedman,andS.D.SarmaZhang,andE.M.Hankiewicz(2010),NewJ.Phys.,12,065012.(2008),Rev.Mod.Phys.,80,1083.RoushanP.,J.Seo,C.V.Parker,Y.S.Hor,D.Hsieh,D.Qian,NielsenH.B.,andM.Ninomiya(1981),Nucl.Phys.B,185,20.A.Richardella,M.Z.Hasan,R.J.Cava,andA.Yazdani(2009),NiemiA.J.,andG.W.Semenoff(1983),Phys.Rev.Lett.,51,2077.Nature,460,1106.NishideA.,Y.Takeichi,T.Okuda,A.A.Taskin,T.Hirahara,RoyR.(2008),e-printarXiv:0803.2868.K.Nakatsuji,F.Komori,A.Kakizaki,Y.Ando,andI.MatsudaRoyR.(2009a),Phys.Rev.B,79,195322.(2010),NewJ.Phys.,12,065011.RoyR.(2009b),Phys.Rev.B,79,195321.NomuraK.,M.Koshino,andS.Ryu(2007),Phys.Rev.Lett.,99,RyuS.,A.P.Schnyder,A.Furusaki,andA.W.W.Ludwig(2010),146806.NewJ.Phys.,12,065010.NomuraK.,andN.Nagaosa(2010),e-printarXiv:1006.4217.SakamotoY.,T.Hirahara,H.Miyazaki,S.-I.Kimura,andNomuraK.,S.Ryu,M.Koshino,C.Mudry,andA.Furusaki(2008),S.Hasegawa(2010),Phys.Rev.B,81,165432.Phys.Rev.Lett.,100,246806.SatoK.,D.Loss,andY.Tserkovnyak(2010),e-printNovikE.G.,A.Pfeuffer-Jeschke,T.Jungwirth,V.Latussek,C.R.arXiv:1003.4316.Becker,G.Landwehr,H.Buhmann,andL.W.MolenkampSauJ.D.,R.M.Lutchyn,S.Tewari,andS.DasSarma(2010),Phys.(2005),Phys.Rev.B,72,035321.Rev.Lett.,104,040502.NovikE.G.,P.Recher,E.M.Hankiewicz,andB.Trauzettel(2010),SchmidtM.J.,E.G.Novik,M.Kindermann,andB.TrauzettelPhys.Rev.B,81,241303.(2009),Phys.Rev.B,79,241306.ObuseH.,A.Furusaki,S.Ryu,andC.Mudry(2008),Phys.Rev.B,SchnyderA.P.,S.Ryu,A.Furusaki,andA.W.W.Ludwig(2008),78,115301.Phys.Rev.B,78,195125.OlshanetskyE.,Z.Kvon,G.Gusev,N.Mikhailov,S.Dvoretsky,andSemenoffG.W.(1984),Phys.Rev.Lett.,53,2449.J.Portal(2010),JETPLett.,91,347.SeoJ.,P.Roushan,H.Beidenkopf,Y.S.Hor,R.J.Cava,andA.Yaz-OregY.,G.Refael,andF.vonOppen(2010),e-printdani(2010),Nature,466,343.arXiv:1003.1145.SeradjehB.,J.E.Moore,andM.Franz(2009),Phys.Rev.Lett.,103,066402. 54ShahilK.M.F.,M.Z.Hossain,D.Teweldebrhan,andA.A.Ba-WimmerM.,A.R.Akhmerov,M.V.Medvedyeva,J.Tworzydło,landin(2010),Appl.Phys.Lett.,96,153103.andC.W.J.Beenakker(2010),Phys.Rev.Lett.,105(4),046803.ShengD.N.,Z.Y.Weng,L.Sheng,andF.D.M.Haldane(2006),WinklerR.(2003),Spin-OrbitCouplingEffectsinTwo-DimensionalPhys.Rev.Lett.,97,036808.ElectronandHoleSystems(Springer).ShindouR.,andS.Murakami(2009),Phys.Rev.B,79,045321.WittelK.,andR.Manne(1974),Theoret.Chim.Acta(Berl.),33,ShitadeA.,H.Katsura,J.Kuneˇs,X.-L.Qi,S.-C.Zhang,andN.Na-347.gaosa(2009),Phys.Rev.Lett.,102,256403.WittenE.(1979),Phys.Lett.B,86,283.SinovaJ.,D.Culcer,Q.Niu,N.A.Sinitsyn,T.Jungwirth,andA.H.WittenE.(1983),Nucl.Phys.B,223,422.MacDonald(2004),Phys.Rev.Lett.,92,126603.WrayL.,S.Xu,J.Xiong,Y.Xia,D.Qian,H.Lin,A.Bansil,Y.Hor,SteinbergH.,D.R.Gardner,Y.S.Lee,andP.Jarillo-Herrero(2010),R.J.Cava,andM.Z.Hasan(2010),unpublished.e-printarXiv:1003.3137.WuC.,B.A.Bernevig,andS.C.Zhang(2006),Phys.Rev.Lett.,StoneM.,C.-K.Chiu,andA.Roy(2010),e-printarXiv:1005.3213.96,106401.Str¨omA.,andH.Johannesson(2009),Phys.Rev.Lett.,102,096806.WuC.,K.Sun,E.Fradkin,andS.-C.Zhang(2007),Phys.Rev.B,Str¨omA.,H.Johannesson,andG.I.Japaridze(2010),Phys.Rev.75,115103.Lett.,104,256804.WuC.,andS.-C.Zhang(2004),Phys.Rev.Lett.,93,036403.SuW.P.,J.R.Schrieffer,andA.J.Heeger(1979),Phys.Rev.Lett.,XiaY.,L.Wray,D.Qian,D.Hsieh,A.Pal,H.Lin,A.Bansil,42,1698.D.Grauer,Y.Hor,R.Cava,andM.Hasan(2009),NaturePhys.,SushkovA.B.,G.S.Jenkins,D.C.Schmadel,N.P.Butch,5,398.J.Paglione,andH.D.Drew(2010),e-printarXiv:1006.1008.XiaY.,L.Wray,D.Qian,D.Hsieh,A.Pal,H.Lin,A.Bansil,SwingleB.,M.Barkeshli,J.McGreevy,andT.Senthil(2010),e-D.Grauer,Y.S.Hor,R.J.Cava,andM.Z.Hasan(2008),e-printprintarXiv:1005.1076.arXiv:0812.2078.TanakaY.,T.Yokoyama,andN.Nagaosa(2009),e-printXuC.,andJ.Moore(2006),Phys.Rev.B,73,045322.arXiv:0907.2088.YanB.,C.-X.Liu,H.Zhang,C.Y.Yam,X.L.Qi,T.Frauenheim,TangH.,D.Liang,R.L.J.Qiu,andX.P.A.Gao(2010),e-printandS.C.Zhang(2010),e-printarXiv:1003.0074.arXiv:1003.6099.YangB.-J.,andY.B.Kim(2010),e-printarXiv:1004.4630.TaskinA.A.,andY.Ando(2009),Phys.Rev.B,80,085303.YaoY.,F.Ye,X.-L.Qi,S.-C.Zhang,andZ.Fang(2007),Phys.Rev.TeoJ.C.,andC.Kane(2010),e-printarXiv:1006.0690.B,75,041401.TeoJ.C.Y.,L.Fu,andC.L.Kane(2008),Phys.Rev.B,78,045426.YeF.,G.H.Ding,H.Zhai,andZ.B.Su(2010),Euro.Phys.Lett.,TeoJ.C.Y.,andC.L.Kane(2009),Phys.Rev.B,79,235321.90,47001.TeweldebrhanD.,V.Goyal,andA.A.Balandin(2010a),NanoLett.,YoungM.W.,S.S.Lee,andC.Kallin(2008),Phys.Rev.B,78,10,1209.125316.TeweldebrhanD.,V.Goyal,M.Rahman,andA.A.BalandinYuR.,W.Zhang,H.J.Zhang,S.C.Zhang,X.Dai,andZ.Fang(2010b),Appl.Phys.Lett.,96,053107.(2010),e-printarXiv:1002.0946.ThoulessD.J.(1983),Phys.Rev.B,27,6083.ZakJ.(1989),Phys.Rev.Lett.,62,2747.ThoulessD.J.(1998),TopologicalQuantumNumbersinNonrela-ZangJ.,andN.Nagaosa(2010),Phys.Rev.B,81,245125.tivisticPhysics(WorldScientific).ZhangG.,H.Qin,J.Teng,J.Guo,Q.Guo,X.Dai,Z.Fang,andThoulessD.J.,M.Kohmoto,M.P.Nightingale,andM.denNijsK.Wu(2009a),Appl.Phys.Lett.,95,053114.(1982),Phys.Rev.Lett.,49,405.ZhangH.,C.-X.Liu,X.-L.Qi,X.Dai,Z.Fang,andS.-C.ZhangTkachovG.,andE.M.Hankiewicz(2010),Phys.Rev.Lett.,104,(2009b),NaturePhys.,5,438.166803.ZhangH.,C.-X.Liu,X.-L.Qi,X.-Y.Deng,X.Dai,S.-C.Zhang,andTranM.-T.,andK.-S.Kim(2010),e-printarXiv:1006.3208.Z.Fang(2009c),Phys.RevB,80,085307.TseW.-K.,andA.H.MacDonald(2010),e-printarXiv:1003.2260.ZhangL.B.,K.Chang,X.C.Xie,H.Buhmann,andL.W.VolkovB.A.,andO.A.Pankratov(1985),JETPLett.,42,178.Molenkamp(2009d),e-printarXiv:0912.3327.VollhardtD.,andP.W¨olfle(1990),TheSuperfluidPhasesofHeliumZhangS.C.(1992),Int.J.Mod.Phys.B,6,25.3(TaylorandFrancis,USA).ZhangS.C.(2008),Physics,1,6.VolovikG.E.(1988),Zh.Eksp.Teor.Fiz.94,123[Sov.Phys.JETPZhangS.C.,andJ.P.Hu(2001),Science,294,823.67,1804].ZhangT.,P.Cheng,X.Chen,J.-F.Jia,X.Ma,K.He,L.Wang,VolovikG.E.(1999),PismaZh.Eksp.Teor.Fiz.,70,601[JETPLett.H.Zhang,X.Dai,Z.Fang,X.Xie,andQ.-K.Xue(2009e),Phys.70,609(1999)].Rev.Lett.,103,266803.VolovikG.E.(2002),JETPLett.,75,63.ZhangW.,R.Yu,H.Zhang,X.Dai,andZ.Fang(2010),e-printVolovikG.E.(2003),TheUniverseinaHeliumDroplet(OxfordarXiv:1003.5082.UniversityPress,Oxford).ZhangY.,K.He,C.-Z.Chang,C.-L.Song,L.Wang,X.Chen,J.Jia,WanX.,A.Turner,A.Vishwanath,andS.Y.Savrasov(2010),e-Z.Fang,X.Dai,W.-Y.Shan,S.-Q.Shen,Q.Niu,X.Qi,S.-C.printarXiv:1007.0016.Zhang,X.Ma,andQ.-K.Xue(2009f),e-printarXiv:0911.3706.WangZ.,T.Lin,P.Wei,X.Liu,R.Dumas,K.Liu,andJ.ShiZhangY.,Y.Ran,andA.Vishwanath(2009g),Phys.Rev.B,79,(2010a),e-printarXiv:1007.0470.245331.WangZ.,X.-L.Qi,andS.-C.Zhang(2010b),e-printZhangY.,andA.Vishwanath(2010),e-printarXiv:1005.3542.arXiv:1004.4229.ZhouB.,H.-Z.Lu,R.-L.Chu,S.-Q.Shen,andQ.Niu(2008),Phys.WangZ.,X.-L.Qi,andS.-C.Zhang(2010c),NewJ.Phys.,12,Rev.Lett.,101,246807.065007.ZhouX.,C.Fang,W.-F.Tsai,andJ.Hu(2009),Phys.Rev.B,80,WeeksC.,andM.Franz(2010),Phys.Rev.B,81,085105.245317.WilczekF.(1987),Phys.Rev.Lett.,58,1799.ZitkoR.(2010),e-printarXiv:1003.5581.WilczekF.(2009),Nature,458,129.ZyuzinV.A.,andG.A.Fiete(2010),e-printarXiv:1006.0498.