D Chong XiaoGang Wen and Marin Solja Department of Physics Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA Received 28 March 2008 revised manuscript received 6 May 2008 published 30 June 2008 We present an effective theory fo ID: 74930
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EffectivetheoryofquadraticdegeneraciesY.D.Chong,Xiao-GangWen,andMarinSoljaDepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139,USAReceived28March2008;revisedmanuscriptreceived6May2008;published30June2008WepresentaneffectivetheoryfortheBlochfunctionsofatwo-dimensionalsquarelatticenearaquadraticdegeneracypoint.Thedegeneracyisprotectedbythesymmetriesofthecrystal,andbreakingthesesymmetries DiracHamiltonianneareachdegeneracypointorDirac.Furthermore,playstheroleofamassterm,open-ingabandgap.Setting0distortstheDiracHamiltoniananditseigenvaluespectrum;forexample,theDiracconesinthe=0limitbecometiltedinspace,asshowninFig..Alongtheline,thesplittingofthedegeneracypointcanbethoughtofasaverticalrelativedisplacementofthetwoparabolicbandsnote,however,thatthebandsmeetonlyatisolatedpointsinthefullThesplittingisaccompaniedbyachangeinthedensityofstatesfromadiscontinuitytoalinearÒdipÓcenteredatthefrequencyofthebanddegeneracy.When0,thedensitystatesisdiscontinuousatthebandedges,droppingtozeroinsidethebandgap.Thesituationisverysimilarfor0.When=0,thedegeneracysplitsintotwobutalongthelineinsteadof.Whenbotharenonzero,thedegeneraciesarelocatedatanintermediatelocation,,sinwheretan,andexpandingaroundeachpointyieldsaDirac-typeHamiltoniananalogoustoEq.0,thebandsarenondegenerate,andtheirChernnumberscanbecalculated.Thedetailsofthiscalcu-lationaregivenintheAppendix,andtheresultisthattheupperandlowerbandspossessChernnumberssgn,respectively,regardlessofthevaluesof.Thisimpliestheexistenceofasinglefamilyofone-wayedgemodesandagreesexactlywiththenumericalre-sultsofWangetal.AlthoughtheeffectiveHamiltonianisonlyvalidnear=0,ityieldsthesameChernnumberastheactualbandstructurebecauseonlytheregionnearthebrokendegeneracypointprovidesanonvanishingÒBerryßuxÓcontributiontotheChernnumber.whileourtheoryonlydescribesweaksymmetrybreaking,theChernnumberisatopologicalquantityandcannotbealteredbynonperturbativedistortionsaslongasthebandsremainnondegenerate,whichiswhyitremainsunchangedeveninthestrongparity-breakingregimeexploredbyWang FIG.2.Coloronlinenicbandstructuresfortime-reversiblelatticesofdielectricrodswithradiusandpermittiv-,plottedagainstthedisplacementfromthecorneroftheBrillouinzone,,alongtheline.Thedotsshownu-mericaldata;thesolidlinesshowtheanalyticresultsaf-terÞttingthefreeparametersintothenumericaldata.Notethatthebandstructureisindepen-dentoftheparameteralongthisgivenline.In,wetake=0.25=16.26,withÞtted0.For=0;for=1.2radwithÞttedvalue=3.7.In,we=0.2=9.92,withÞttedvalue0.5.In=0;for=10withÞttedvalue=1.8.In,athreedi-mensionalplotofthebandstruc-tureofsystemisshown.EFFECTIVETHEORYOFQUADRATICDEGENERACIESPHYSICALREVIEWB,235125 ;thecalculationfortheupperbandproceedsanalo-gously.First,consider=0.WenotethattheeigenvectorsoftheeffectiveHamiltoniandonotdependonsincethatparametermultipliestheidentitymatrix.Forsimplicity,we=1.Theeigenvectorcorrespondingtothelowerbandis 2 4+22+2 sin2cos2sin2 regardlessofthevaluesof.Here,isthecylindricalcoordinaterepresentationof.TheBerryconnectioniscos2sin2 4+22 sin2ToobtaintheChernnumber,weintegratetheBerryconnec-tionaroundaloop 2i=0dáA2 sin2 0+2+0 sin2Theintegralcanbeperformedviathesubstitutionsin2=tanh,andweobtain=sgn Asdiscussedinthetext,theaboveresultremainsunchangedevenwhenweenterthenonperturbativeregime,eventhoughoureffectivetheoryisonlyvalidforsmallvaluesofarenonzero,thebandmaximumat=0splitsintotwodistinctmaxima,andexpandingaroundeachmaximumyieldsaDirac-typeHamiltonian.Forin-stance,when=0and0themaximaoccurat 3/ ,andtheHamiltonianneareachofthesepointsisgivenbyEq..IntermsofthevariablesdeÞnedinEq.,theBerryconnectionforthelowerbandis +sin q2+b2 q q2+b2sin,whereb2/ referstowhichmaximumweareex-pandingaround,andisthecylindricalcoordinaterep-resentationof.ThisBerryconnectionhasthesameformasbutwindshalfasquicklyaroundeachmaximumpointasEq.doesaround=0.Eachmaximumthuscontributessgn2totheChernnumberofthelower Z.Wang,Y.D.Chong,J.D.Joannopoulos,andM.SoljaPhys.Rev.Lett.,013905TheQuantumHallEffect,editedbyR.E.PrangeandS.M.Springer-Verlag,NewYork,1987F.D.M.HaldaneandS.Raghu,Phys.Rev.Lett.,013904S.RaghuandF.D.M.Haldane,arXiv:cond-mat/0602501F.D.M.Haldane,Phys.Rev.Lett.,2015B.Simon,Phys.Rev.Lett.,2167Y.Hatsugai,Phys.Rev.Lett.,3697S.G.JohnsonandJ.D.Joannopoulos,Opt.Express,173COMSOLMULTIPHYSICS3.3,COMSOLInc.,www.comsol.comM.PlihalandA.A.Maradudin,Phys.Rev.B,8565CHONG,WEN,ANDSOLJAPHYSICALREVIEWB,235125