拓扑电子态-1pdf

1拓扑量子态与拓扑电子材料方忠中国科学院，物理研究所Acknowledgement:Theory:X.Dai,H.M.Weng,G.Xu(IoP)Exp:YulinChen(Oxford)andZXShen(Stanford)MingShi(PSI),Y.G.Shi,X.J.Zhou,andL.Lu(IoP)2Contents一、简介：对称性，拓扑量子态二、拓扑不变量：三、拓扑量子态（动量空间）:（1）ChernInsulators（2）Z2TopologicalInsulators（3）TopologicalSemimetals（4）TopologicalSuperconductors四、拓扑电子材料：Bi2Se3,Bi2Te3,Cr-Bi2Te3,HgCr2Se4,Na3Bi,Cd3As2,etc.一、简介：对称性“对称性”是物理学研究的重要部分1),对称性：守恒量3),对称性破缺：物质状态改变2),对称性分类：丰富多彩例如，Lorentz不变性：时间平移能量守恒空间平移动量守恒，空间旋转角动量守恒例如：局域与整体对称性，分立与连续对称性Time-Rversal,Parity,Gauge,Particle－Hole,etc（1）直接对称性破缺：Hamiltonian对称性破缺（2）自发对称性破缺：H没有破缺，但是波函数对称性破缺2013诺奖：Higgs机制自发对称性破缺一、简介：凝聚态物质准粒子（低能物理）电子三个量子特征，电荷、自旋、轨道准粒子集体激发，相干性费米统计化学势、费米面晶体能带理论，动量空间多体、多自由度？强关联？宏观量子现象？一、简介：量子有序态有序态是凝聚态物理研究的基本内涵之一例如：磁有序态、电荷有序态、超导态等局域有序态：对称性破缺导致有序态（朗道对称性破缺理论）1.物态可以用局域序参量描写如：铁磁态的磁化强度M(r)2.相变伴随着对称性破缺如：M(r)的出现破坏了旋转对称性整体有序态：拓扑量子态（量子物理与几何的完美结合）1.具有拓扑性质的“量子态”2.不能用局域序参量描写，而要用全局拓扑不变量描写3.相变过程并不伴随对称性破缺拓扑“0”拓扑“1”反铁磁铁磁自旋波vs.Phase非常重要一、简介：拓扑量子态新物态、新奇量子现象除非破坏性剪断拓扑量子态的优点：对细节不敏感1.“0”与“1”严格区分，无微扰过程，不怕干扰、噪声2.与“奇点”密切相关，在边界上会有特殊量子态≠信息高速公路：极低电阻、极低能耗各行其道，永不混杂遇到杂质，自动绕行鱼目混杂，杂乱无章遇到杂质，会被散射普通态拓扑有序态奇点面包圈球过渡一、简介：拓扑态的边界效应CIVaccumNormalinsulatorBoundaryCut:NoadiabaticconnectionbetweentwosidesN=1N=0“twistedband”BoundaryState二、拓扑不变量：晶格平移不变性与动量空间实空间：倒空间（动量空间）：(第一布里渊区)BlochState：GaugeFreedom：abRikakb二、拓扑不变量：Berryconnection&CurvatureK-spaceastheparameterspace:Connection:Curvature:GaugedependentGaugeinvariantSymmetry:ForsystemswithbothTandI二、拓扑不变量：GaugeFieldGaugecovariantpositionoperator:Commutationrelation:Curvaturecanbewritteninanti-symmetrictensorform:istheLevi-Civitaantirsymmetirctensor,and二、拓扑不变量：MagneticFieldinK-spaceKeyquantity:canbeviewedasmagneticfieldink-space[Sundaram&Niu,et.al,PRB(1999);Jungwirth&Niu,et.al,PRL(2002);Fang,Nagaosa,Tokura,et.al,Science(2003);Y.Yao&Niu,et.al.PRL(2004)]nkknkkuui∇×∇=×∇=)k(A)k(ΩkfunctionBlochofpartperiodicconnectionBerrykAk:,:)(nuAnomalousHallEffect二、拓扑不变量：可观测量，HallConductivityB＝0jxEyHallconductivity:For2Dinsulators:fn(k)=1or0二、拓扑不变量：ChernNumberCherntheorem:Z=intergerChernnumber2DBrillouinZone:Quantization:二、拓扑不变量:ChernNumberasWannierCenterBerryPhase:WindingNumerN≠kyWannierCenter:二、拓扑不变量：T-symmetryandZ2InvariantNoTWithTChernInstulatorZ2TITRSZ-ZInvariant:Z2=Zmod2Ref:(1)Hasan&Kane,RMP(2010).(2)Qi&Zhang,RMP(2011).Z2TIcanbeextendedto3D!二、拓扑不变量：MagneticMonopoleandWeylnodeAny2x2Hamiltonian:SimplestCase:(1)TopologicalObjects(2)Gapless,nomassterm(3)Chirality±(leftorright-hand)(4)Protectedbytranslation(kmustbewelldefined)MagneticMonopoles:Q=magneticChargeFang,Science(2003).Weylnodes:NS二、拓扑不变量：TopologicalInvariantfor3DMetalsDefinition:CFS=0,normalmetalCFS≠0,topologicalmetalifEfatnode(k=0)topologicalsemimetalNotes:(1)|Q|canbemorethan1(2)+Q&-Qmonopoleshavetoappearinpairinlattice,butmayseparateinK.(3)+Q&-Qmonopolescanannihilate.(4)Definedonlyfor3Dk-spaceFSVolovik,JETP(2002).X.L.Qi,et.al.,PRB(2010).Z.J.Wang,et.al.,PRB(2012)三、拓扑量子态：MomentumSpace三、拓扑量子态:丰富的拓扑电子态量子自旋霍尔效应2D拓扑绝缘体3D拓扑绝缘体拓扑金属量子霍尔效应Z≠0Z2≠0Z2≠0CFS≠0三、拓扑量子态：IntegerQuantumHallEffectQuantumHallEffect2DEGLandauLevelsBulkstatesEdgestatesEdgestatesQuantizationK.VonKlitzing,et.al.,PRL(1980)典型的宏观量子现象！Dissipationlessedge有电流，没有电压HallEffectρHB三、拓扑量子态：IQHEZ-numberforT-brokenQHorQAHinsulators(2D)WindingNumerNTKNN,PRL(1982):Haldane,PRL(1988):LatticeModel(Honeycomb);Realization:Fang&Dai,Science2010(Theory);Xue,Science2013(Exp.)(Cr-doped(BiSb)Te3thinfilm)n=ChernnumberOnoda&Nagaosa:PRL(2003);S.C.Zhang:PRB(2006);PRL(2008)四、拓扑电子材料:BandInversionMechanismGuidelines:1.Semiconductorwithinvertedbandstructure2.StrongSOC四、拓扑电子材料:First-principlesCalculationsMatrixelement:Berryphase:orWilsonLoopmethodtoavoidthegauge-fixconditionWindingnumber:Bi2Se3四、拓扑电子材料:重要进展2DTI(QSHInsulator):HgTeQuantumWellTheory:Bernevig,Science(2006),Exp:Konig,Science(2007).3DTI:Bi2Se3,Bi2Te3,Sb2Te3Theory:H.J.Zhang,Nat.Phys.(2009),Exp:Y.Xia,Nat.Phys.(2009)Y.L.Chen,Science(2009).QAHEInsulator:Cr-dopedBi2Te3Theory:R.Yu,Science.(2010),Exp:C.Z.Chang,Science(2013).四、拓扑电子材料:QuantumHallTrioHgTeBi2Te3,Bi2Se3withCr-doping四、拓扑电子材料：拓扑Weyl半金属Topologicalphasetransitionasfunctionofkz.Considering2Dsheet(fixedkz=m).3DWeylsemimetal:(1)Spin-MomentumLockin3D(2)Fermiarcsonsidesurface,(3)Magneticmonopolesinbulk,(4)QAHEinQuantum-wellstructure(5)ABJanomaly,andetc.(NegativeMRforE//B)m0,ChernIm0,NIX.G.Wan,et.al.PRB(2011).A.A.Burkov,et.al,PRL(2011);PRB(2011)G.Xu,et.al.,PRL(2011).四、拓扑电子材料：拓扑DiracSemimetalIfbothTandIsymmetryarepresent:+Q&-QWeylnodeshavetooverlapinK-space3DDiracsemimetal:(1)Pseudofermiarcsonsurface(2)Giantdiamagnetism:χ(ε)≈log(1/ε)(3)LinearQuantummagneto-resistance.(4)QSHEinitsquantum-wellstructureNeedcrystalsymmetryprotection.+CaseI:M≠0,InsulatorCaseII:M=0,3DDiracSemimetalS.M.Young,et.al.,PRL(2012).Z.J.Wang,et.al.,PRB(2012)CriticalPoint提出了拓扑金属的定义：PRB85,195320（2012）定义:CFS=0,一般金属CFS≠0,拓扑金属如果费米面正好在能带交叉点拓扑半金属FS(费米面)预言了拓扑半金属材料：（1）HgCr2Se4:PRL107,186806（2011）（2）Na3Bi:PRB85,195320（2012）（3）Cd3As2:PRB88,125427（2013）拓扑半金属具有与普通金属不同的新奇物性：磁阻、抗磁性、拓扑超导等四、拓扑电子材料：拓扑半金属与实验合作，两个理论预言工作都被证实Science343,6173(2014)：Na3BiNatureMaterials,3990(201