凝聚態(tài)物理學(xué)講課內(nèi)容第0章

拓撲能帶理論簡介第一章

（傳統(tǒng)物理學(xué)）概論第二章GeometryPhase(BerryPhase)第三章DiracEquationinCMP第四章ChernInsulators第五章TopologicalInsulators第六章TopologicalSemimetals參考書目馮端,金國鈞,凝聚態(tài)物理學(xué)新論,上海科學(xué)技術(shù)出版社.2.李正中，

固體理論，

高等教育出版社3.P.M.Chaikin&T.C.Lubensky,Principlesofcondensedmatterphysics,Cambridge(1995).P.W.Anderson,Basicnotionsofcondensedmatterphysics,Benjamin-Cummings,MenloPark(1984)5.B.A.Bernevig&T.L.Hughes,TOPOLOGICALINSULATORSANDTOPOLOGICALSUPERCONDUCTORS,(2013)6.Shun-QingShen,

TopologicalInsulatorsDiracEquationinCondensedMatters,Springer(2012)學(xué)習(xí)成績平時成績（66％）＋Project（34％）1.平時成績:作業(yè)（33%）+考勤（33%）2.Project要求：與本課程相關(guān)的，基于閱讀多篇重要文獻以及綜述后的關(guān)于某一新奇效應(yīng)、新物理現(xiàn)象、新物理概念、新材料的奇異物性、相關(guān)理論方法等的較為深入的介紹。第0章

拓撲能帶理論簡介課程的主要內(nèi)容包括：介紹量子霍爾效應(yīng)，量子反?；魻栃?yīng)，拓撲絕緣體（強或弱拓撲），高階拓撲絕緣體，拓撲半金屬（Dirac或者Weyl）以及它們的模型和材料實現(xiàn)。同時講解基本的理論知識，如Berryphase，Chernnumber，windingnumber，Z2number等拓撲不變量，以及拓撲能帶理論等。凝聚態(tài)物理學(xué)進入拓撲+的新時代！第0章

拓撲能帶理論簡介拓撲量子計算低功耗自旋電子學(xué)新型紅外光電探測拓撲物理研究的重要應(yīng)用價值拓撲物理研究的重大科學(xué)意義全新的物態(tài)，超越了朗道范式2016年諾貝爾物理學(xué)獎第一章傳統(tǒng)的凝聚態(tài)理論重要概念:

對稱破缺，序參數(shù)和元激發(fā)凝聚態(tài)物理的兩塊基石：朗道費米液體理論和朗道對稱性破缺理論對稱性對稱性（symmetry）是現(xiàn)代物理學(xué)中的一個核心概念，系統(tǒng)從一個狀態(tài)變換到另一個狀態(tài)，如果這兩個狀態(tài)等價，則說系統(tǒng)對這一變換是對稱的?；蛘哒f給系統(tǒng)一個“操作”，如果系統(tǒng)從一個狀態(tài)變到另一個等價的狀態(tài)，則說系統(tǒng)對這一操作是對稱的。它泛指規(guī)范對稱性（gaugesymmetry，或局域?qū)ΨQ性localsymmetry）和整體對稱性（globalsymmetry）。它是指一個理論的拉格朗日量或運動方程在某些變量的變化下的不變性。如果這些變量隨時空變化，這個不變性被稱為規(guī)范對稱性，反之則被稱為整體對稱性。物理學(xué)中最簡單的對稱性例子是牛頓運動方程的伽利略變換不變性和麥克斯韋方程的洛倫茲變換不變性。數(shù)學(xué)上,利用群論來研究對稱性。對稱性的性質(zhì)對稱性可以是分離的(具有有限的數(shù)目)例如:八面體分子的轉(zhuǎn)動、分子的轉(zhuǎn)動與反射、晶格的平移。也可以是連續(xù)的(具有無限的數(shù)目)例如:原子或核子的轉(zhuǎn)動。對稱性可以是更一般的和抽象的，如與規(guī)范理論相關(guān)的對稱性。對稱性分類自然界的四類對稱性:(1)全同粒子的互換(2)連續(xù)時空變換,如平移,旋轉(zhuǎn)和加速

(3)分立變換,如空間反演,時間反演,粒子-反粒子共軛(4)規(guī)范變換,如U(1)(電荷,超荷,重子數(shù)和輕子數(shù)守恒),SU(2)(同位旋)

和SU(3)(色和味)對稱

對稱性都是植根于某些物理量是不可觀測的假設(shè),不可觀測量存在的直接后果是出現(xiàn)守恒律或選擇定則.相反,一旦一個不可觀測量變成可觀測的,對稱性就破缺了.對稱性與守恒定律物理系統(tǒng)的每一個對稱性都有相對的守恒定律--諾特定理。反過來說：物理系統(tǒng)有某守恒性質(zhì)就代表它具有相應(yīng)的對稱性。例如，空間位移對稱造成動量守恒。而時間平移對稱造成能量守恒：為何過去和現(xiàn)在事物運動的規(guī)律是相同的？那是因為運動規(guī)律在時間平移的變動中能夠保持不變。對稱破缺對稱性破缺（symmetrybreaking）系指物理學(xué)里，在具有某種對稱性的物理系統(tǒng)之臨界點附近發(fā)生的微小振蕩，通過選擇所有可能分岔中的一個分岔，打破了這物理系統(tǒng)的對稱性，并且決定了這物理系統(tǒng)的命運。

自發(fā)對稱性破缺自發(fā)對稱性破缺（spontaneoussymmetrybreaking）描述物理系統(tǒng)的拉格朗日量或哈密頓量具有某種對稱性，但是物理系統(tǒng)的最低能量態(tài)（真空態(tài)）不具有此種對稱性。通常，這種對稱性破缺會具有一種有序參數(shù)。自發(fā)對稱性破缺對稱破缺與Goldstone定理與對稱性破缺相關(guān)的一個結(jié)論是Goldstone定理:它是指在具有連續(xù)對稱性破缺的相對論量子場論中必然存在無質(zhì)量的粒子-Goldstone玻色子。在固體理論中,Goldstone玻色子是集團激發(fā)的聲子。晶體只有離散的平移對稱性，破缺了連續(xù)的平移對稱性。周光召先生:“對稱性和對稱破缺是世界統(tǒng)一性和多樣性的根源”生命起源中的對稱破缺：DNA左右鏡像對稱破缺！A-DNA、B-DNAZ-DNA右手雙螺旋左手雙螺旋藝術(shù)中的對稱性破缺維納斯女神《向日葵》梵高1888（荷蘭）自然界中的對稱破缺

凝聚態(tài)中的對稱性破缺凝聚態(tài)物質(zhì)世界大多數(shù)是對稱破缺的產(chǎn)物:晶體是平移對稱破缺的產(chǎn)物(原子位置的周期性破壞了任意平移的不變性);空間反演對稱性的破缺產(chǎn)生了鐵電體;時間反演對稱性的破缺產(chǎn)生磁有序結(jié)構(gòu);規(guī)范對稱性的破缺產(chǎn)生了超流體與超導(dǎo)電體...序參數(shù)序參量:

低溫有序相的一個標志,描述偏離對稱的性質(zhì)和程度.為某個物理量的平均值,可以是標量,矢量,復(fù)數(shù)或更加復(fù)雜的量，是一個局域的量.隨對稱性的不同,它在高溫時為零,而低溫下取有限值,在Tc處轉(zhuǎn)變.對稱破缺意味著序參量不為零的有序相的出現(xiàn).相變和臨界現(xiàn)象相變:

定義：一個多粒子系統(tǒng)在不同的溫度和壓強或其他外部條件下可以處在不同的狀態(tài),不同狀態(tài)之間的轉(zhuǎn)變叫相變.相變的分類標志:熱力學(xué)勢及其導(dǎo)數(shù)的連續(xù)性.熱力學(xué)勢:自由能,內(nèi)能

一階導(dǎo)數(shù):壓力(體積),熵(溫度),平均磁化強度等二階導(dǎo)數(shù):壓縮系數(shù),膨脹系數(shù),比熱,磁化率等.一級相變或不連續(xù)相變:

熱力學(xué)勢連續(xù),一階導(dǎo)數(shù)不連續(xù)的狀態(tài)突變二級相變或連續(xù)相變:熱力學(xué)勢和一階導(dǎo)數(shù)連續(xù),二階導(dǎo)

數(shù)不連續(xù)的狀態(tài)突變連續(xù)相變理論：平均場理論（唯象理論）

平均場理論:被多次發(fā)明的理論1873:vandeWaals氣液狀態(tài)方程1907:Wiess鐵磁相變的“分子場理論”1934:二元合金有序-無序轉(zhuǎn)變的Bragg-Williams近似1937:Landau相變理論Landau的二級相變理論Landau的二級相變理論:

強調(diào)對稱性的重要性,對稱性的存在與否是不容模棱兩可的，高對稱性相中某一對稱元素突然消失，就對應(yīng)于相變的發(fā)生，導(dǎo)致低對稱相的出現(xiàn)。核心：對稱破缺特例：連續(xù)相變不存在對稱性上的差別(汽-液相變)對于沒有破缺對稱性的系統(tǒng)，應(yīng)選取某個對相變點上下兩相之間的差別敏感的量與它在相變點的差別為序參量。

自由能作為序參量的函數(shù)。序參量：標量、矢量、張量或復(fù)數(shù)。

：矢量，在相變點，將自由能展開：不含奇次冪項要求：高于相變溫度時，

＝0使系統(tǒng)自由能達到極??；低于相變溫度時，

，使系統(tǒng)自由能達到極小。Landau的二級相變理論－Formula能取到極小值表明：(2)因子

使自由能達到極小，使自由能達到極小，連續(xù)變化要求，(3)有序和無序：將自由能F對

取極小是出現(xiàn)極小值的唯一解，對應(yīng)無序態(tài)!F上述方程有非零解,對應(yīng)有序態(tài)!

F(4)λ點均為溫度的緩變函數(shù),(不參與求導(dǎo))比熱在相變溫度點不連續(xù)：傳統(tǒng)凝聚態(tài)物理中的對稱破缺現(xiàn)象LandauFermiLiquidTheoryFermi氣體：

均勻的無相互作用的自由電子氣。較強關(guān)聯(lián)下，電子系統(tǒng)被稱為電子液體或費米液體或Luttinger液體(1D)

費米溫度：均勻的無相互作用的三維系統(tǒng)，費米溫度：費米簡并系統(tǒng)：費米子系統(tǒng)的溫度通常遠低于費米溫度室溫下金屬中的傳導(dǎo)電子LandauFermiLiquidTheoryThekeyideasbehindLandau'stheoryarethenotionofadiabaticityandtheexclusionprinciple.Consideranon-interactingfermionsystem(aFermigas),andsupposewe"turnon"theinteractionslowly.Landauarguedthatinthissituation,thegroundstateoftheFermigaswouldadiabaticallytransformintothegroundstateoftheinteractingsystem.ByPauli'sexclusionprinciple,thegroundstateofaFermigasconsistsoffermionsoccupyingallmomentumstatescorrespondingtomomentump<pFwithallhighermomentumstatesunoccupied.Asinteractionisturnedon,thespin,chargeandmomentumofthefermionscorrespondingtotheoccupiedstatesremainunchanged,whiletheirdynamicalproperties,suchastheirmass,magneticmomentetc.arerenormalizedtonewvalues.Thus,thereisaone-to-onecorrespondencebetweentheelementaryexcitationsofaFermigassystemandaFermiliquidsystem.InthecontextofFermiliquids,theseexcitationsarecalled"quasi-particles”.朗道費米液體理論：

單電子圖象不是一個正確的出發(fā)點，但只要把電子改成準粒子或準電子，就能描述費米液體。準粒子遵從費米統(tǒng)計，準粒子數(shù)守恒，因而費米面包含的體積不發(fā)生變化。假設(shè)激發(fā)態(tài)用動量

表示朗道費米液體理論的適用條件：(1).必須有可明確定義的費米面存在(2).準粒子有足夠長的壽命朗道費米液體理論是處理相互作用費米子體系的唯象理論。在相互作用不是很強時，理論對三維液體正確。二維情況下，多大程度上成立不知道。一維情況下，不成立。Luttinger液體一維：低能激發(fā)為自旋為1/2的電中性自旋子和無自旋電荷為

的波色子的激發(fā)。

Luttinger液體非費米液體行為：與費米液體理論預(yù)言相偏離的性質(zhì)。凝聚態(tài)中的新發(fā)展量子霍爾效應(yīng)（IQH,FQH）的發(fā)現(xiàn)完全出乎人們的意料，揭開了凝聚態(tài)發(fā)展的新篇章。這些新奇的量子態(tài)在零溫或者低溫時包含了許多對稱性相同而本質(zhì)又不同的態(tài)。所以這些相就不能用對稱性加以區(qū)分，故也不能用朗道對稱破缺理論描述。這時，描述體系的相更多的依賴整體的性質(zhì)，而不是local的序參量。引入了拓撲的概念。如symmetryprotectedphases（SPT），包括拓撲絕緣體，拓撲超導(dǎo)體，拓撲半金屬等。電子在電磁場中的運動（經(jīng)典）運動方程等式左邊第一項是加速度項，第二項是碰撞項；右邊是電子受到的Lorenz力。當(dāng)磁場B平行于z軸時，上述運動方程如下：對于靜電場中的穩(wěn)態(tài)，時間導(dǎo)數(shù)為0，于是漂移速度為回旋共振頻率?；魻栃?yīng)

（Halleffect）

byEdwinHallin1879當(dāng)施加了外磁場B的導(dǎo)體中通過傳導(dǎo)電流j時將會產(chǎn)生橫跨導(dǎo)體兩個面的電場，其方向為jxB，該電場稱為Hall電場。如右圖，x方向的電流，z方向的磁場。y方向不能傳導(dǎo)電流，則vy＝0。則有如下橫向（Hall）電場：漂移速度剛建立漂移速度穩(wěn)恒所謂的Hall系數(shù)如下定義：所謂的Hall系數(shù)如下定義，并利用jx=ne2tauEx/m：對于電子，取負號?？捎糜谳d流子濃度測量，種類測定（hore？）霍爾效應(yīng)

（Halleffect）利用上面的漂移速度公式，可以得到如下的靜態(tài)電流密度表達式：其中的系數(shù)即為靜態(tài)磁致電導(dǎo)率張量。在強磁場下，wc*tau>>1，Hall電導(dǎo)率如下此時，Hall電導(dǎo)率反比于磁場。磁導(dǎo)率為（近）自由電子在磁場中的量子理論（近）自由電子在磁場中的量子理論（近）自由電子在磁場中的量子理論（近）自由電子在磁場中的量子理論（近）自由電子在磁場中的量子理論（近）自由電子在磁場中的量子理論（近）自由電子在磁場中的量子理論（近）自由電子在磁場中的量子理論量子霍爾效應(yīng)

（QHE）

byKlausvonKlitzingin1980在低溫、強磁場、高樣品質(zhì)量B電導(dǎo)呈現(xiàn)極其高精度的整數(shù)平臺。1.thevonKlitzingconstantRK=h/e2=25812.807557(18)Ω.被用作電阻標準。2.用于測量精細結(jié)構(gòu)常數(shù)e2/hc.3.要解釋為什么會有如此高

精度的量子化電導(dǎo)平臺（即使

在有一些雜質(zhì)的情況下），這就涉及了本

課程的重點：幾何相位，拓撲能帶理論的

知識。播放動畫Geometricangleandparalleltransportvneverrotatesaboute3paralleltransport:Wavefunctionparalleltransport:Gaugecovariantderivative:Geometricangle和傅科擺Paralleltransportofavectoraroundaclosedloop(fromAtoNtoBandbacktoA)onthesphere.Theanglebywhichittwists,\alpha,isproportionaltotheareainsidetheloop.傅科擺傅科擺擺動平面偏轉(zhuǎn)的角度：φ代表當(dāng)?shù)氐乩砭暥?，t為偏轉(zhuǎn)所用的時間，用小時作單位θ°=15°*t*sinφ播放動畫Berryphase:GeneralformalismQ` `23454rSeveralremarks:Equation(2.12)includesonlytheeigenstate|n>anditsderivatives,butEquation(2.15)showsthattheBerrycurvaturecanbethoughtofastheresultoftheinteractionwiththelevel|n>oftheotherlevels|m>thathavebeenprojectedoutbytheadiabaticinteraction.IfwesumovertheBerryphaseofallenergylevels,weget0,showingthatthesumofallbandscanhaveonlyzeroBerryphase

(Homework).ThecurrentformalismisvalidforthecasewherethelevelEn

issinglydegenerate.Fordegenerateenergylevels,theBerryvectorpotentialbecomesamatrixofdimensionequaltothedegeneracyofthelevels—itbecomesnon-Abelian.OneofthemostimportantapplicationsoftheBerryphaseistheclassificationofdegeneracies.Thiswillbeoneofthemainingredientsofbandcrossingsintopologicalbandtheory.IfthedenominatorofEquation(2.15)isclosetozero,wenowshowthatthisleveldegeneracypointcorrespondstoamonopoleintheparameterspace.Ifthereisamonopole,whataretheformsoftheMaxwellequations?(Homework)Example:two-levelsystemTwo-LevelSystemUsingtheBerryCurvatureGeneralcase1Two-LevelSystemUsingtheHamiltonianApproachIfCenclosethemonopole,Berryphase=+/-2*pi;IfCdoesnot,Berryphase=0.SpininaMagneticFieldTheBerryphaseisthefluxthroughtheareaboundedbyCofamonopoleofstrength?nlocatedattheoriginofthedegeneracy.TheBerryphaseisequaltontimesthesolidanglethattheclosedcontourCsubtendsatB=0.Forhalf-integerspinfermions(n=(2m+1)/2),awholeturnofB(i.e.,arotationthrough2p,inaplane)givesX=2p,whichinturngivesexp(icn(C))=?1.Hence,forhalf-integerspinfermions,thesignchangeofspinorsfroma2*pirotationandthesignchangeofwavefunctionsaroundadegeneracypointofatwo-levelsystemhaveidenticalorigin.CantheBerryPhaseBeMeasured?Nophysicalpropertyisexperimentallyinterestingifitcannotbemeasured.Experimentalsetup:Splitabeamofparticles,allpreparedinadefinitespinstatenintwopaths.Ononepath,Bisconstant,whereasontheotherpathBisconstantinmagnitude,butitsdirectionslowlyvariesaroundaclosedpathCsubtendingasolidangleAfterpassingthroughthisfieldconfiguration,thetwobeamsarecombinedatdetector.ThedynamicalphasefactorisidenticalbetweenthetwobeamsbecausetheenergyEn(B)dependsonlyonthemagnitudeofBwhichisthesame.ThebeamthathasundergonetheBchangeacquiresaBerryphase.TheintensityofthediffractionpatternswillbeTheintensityvariationcanbemeasuredasthemagneticfieldisslowlyvariedtoundergothepathC.CantheBerryPhaseBeMeasured?

ABeffectAnomalousvelocityEffectivedynamicsofBlochelectron

DualitybetweenRealandMomentumSpacesk-spacecurvaturer-spacecurvatureTimereversalsymmetrySpinlesscaseTimeReversalinCrystalsforSpinlessParticlesSpinfulcaseKramers’Theorem=backscatteringisforbiddenTime-ReversalSymmetryinCrystalsforHalf-IntegerSpinParticlesVanishingofBerryPhase(HallConductance)forT-InvariantFermionsBerrycurvature:oddfunctionThe

NobelPhysicslaureates

in1933lineardependenceofmomentumSquaredependenceofmomentum一般認為：前者適用于固體材料，后者適用于高能物理。真是這樣的嗎？The

NobelPhysicslaureates

in1933lineardependenceofmomentumSquaredependenceofmomentum一般認為：前者適用于固體材料，后者適用于高能物理。No！2DmasslessDiracequationingrapheneThelow-energyeffectivemodelis2DmasslessDiracequation!Dirac

comes

back！不僅graphene的低能物理遵循Dirac方程；其它二維材料，如硅烯、鍺烯、錫烯；最近凝聚態(tài)物理出現(xiàn)的拓撲絕緣體；拓撲半金屬…這些都需要用Dirac方程來描述！ReviewofDiracEquationinEnergyspectrumByTaylorexpansionofthedispersionrelationE=root(m2c4+p2c2)≈mc2+p2/2mforsmallmomentump(i.e.,low-energylimit),wecangettheSchrodingerEquation.DiracequationinlowdimensionForthe3DDiracequation,aswehaveshown,itisnecessaryforαandβmatricestobe4×4matrixbecausetherearetotallyfourmatriceswhichanti-commutatewitheachother;For2×2Paulimatrices,themaximumnumberis3;FortheDiracequationinlowerdimension(2Dor1D),itispossibletoconstructDiracequationjustbythreePaulimatrices.DiracequationinCMPOriginally,Diracequationisutilizedtodescribetheevolutionoftheelectroninthevacuumwithquitehighkineticenergy(orlargemomentumpc～mc2).Inthecondensedmatterphysics,theenergyscaleisrelativelylowandtheevolutionoftheelectroncanbedescribedbySchrodingerequation,whichisinfactalowenergyeffectivetheoryofDiracequation.ThiscanbeeasilyseenfromtheTaylorexpansionofthedispersionrelationE=root(m2c4+p2c2)≈mc2+p2/2mforsmallmomentump.Surprisingly,whenwegotoevenlowerenergyafterintegratingouttheinfluencefromthelatticeenvironment,theeffectivetheoryoftheelectronisnotnecessarytobedescribedbySchrodingerequation.Instead,Diracequationmaycomeback!Spin-orbitCouplingSpin-OrbitCouplingOrigin:‘‘Relativistic’’effectinatomic,crystal,impurityorgateelectricfield=Momentum-dependentmagneticfieldStrengthtunableincertainsituationsTheoreticalIssues:ConsequencesofSOCinvarioussituations?InterplaybetweenSOCandotherinteractions?Practicalchallenge:ExploitSOCtogenerate,

manipulateandtransportspinsTight-bindingmodel

緊束縛模型Tight-bindingmodelSeveraladvantages:simpleandtransparent,canbestartedfromtheverybasiclevel;naturallytakingintoaccountboththelatticeenvironmentandthesymmetryoftheatomicorbitals,soalltheessentialphysicscanbeeasilycaptured;givesanaturalcut-offinthehighenergylevel.Slater-KostermethodinthesecondquantizationlanguageDiracequationin1Dsystem:spmodelandSSHmodel2DmasslessDiracequationingrapheneSolid-Statebook:Amaterial

withbandsfullyfilledorwithoutFermisurface.(onlyforbandinsulators).Kohn:Amaterialforwhichallelectronicphenomenaarelocal,i.e.insensitivetoboundaryconditions.(forbandinsulators,Mottinsulator,Andersoninsulators,etc.)Kohn,Phys.Rev.133A171(1964)Whatisaninsulator?Fromnormaltotopologicalinsulator整數(shù)量子霍爾效應(yīng)See

movieGauss-Bonnet-Chern公式TKNN(Chernnumber)&Hallconductance

BerryPhase量子反?；魻栃?yīng)(QAH)

HaldaneModelTopologicalinvariants--Chernnumber

inT-symmetrybreakingsystemChernnumbermustbeaninteger,equalsthenumberofmonopolesinsidethetorus.ItisareflectionofthefactthatasmoothgaugechoiceisNOTpossibleovertheentireBZ.ChernInsulatorTheHallconductanceequalstheChernnumberandisanintegeronlyifthebasemanifold(theBZ)iscompact.Inthecontinuum,themomentumrunsoveranoncompactmanifold(theinfiniteEuclideanplane),andthisdoesnotapply.ZeromodeinDiracequationandsurfacestateintopologicalinsulator0m0-m00m0-m0---m0-m0SomeCommentsontheedgestatesTheedgemodeofthequantumanomalousHalleffectisveryspecial,sinceitonlypropagatesalongonedirection.Suchtypeofedgemodeiscalledchiraledgestate.Itcanalsobeunderstoodasakindoffractionalization.AsshowninFig.(a),thenormal1Dsystemalwayshastwobranches,oneleftmoverandonerightmover.WhenweconsideraquantumanomalousHallinsulatorwithtwoedges,wewillseethatthetwobranchesarespaticallyseparatedintotwooppositeedges,thereforeitcanbeviewedasafractionalizationofthenormal1Dsystem.Alsoduetothespatialseparationofthetwobranches,theycannotbescatteredintoeachotherbyanylocalperturbation,suchasdisorder.Whentheelectronintheedgemodeencounteranimpurity,itwillgoaroundtheimpurityinsteadofthebackscatteringsinceLOCALLYthereisnomodewhichcanbebackscatteredinto.量子反?；魻栃?yīng)

來自中國科學(xué)家的貢獻量子反常霍爾效應(yīng)的實驗實現(xiàn)Time-Reversal-InvariantTopologicalInsulatorsTheHaldanemodelofaCherninsulatorshowsthatanontrivialinsulatorwithnonzeroHallconductancecanexistwhenTRsymmetryisbroken.Morethan15yearsaftertheHaldanemodelwaspublished,itwasrealizedthatkeepingsymmetriesintactgivesrisetosystemsasinterestingastheoneswheresymmetriesarebroken.KaneandMelefirstrealizedthatbydoublingtheHaldanemodeloftheCherninsulatorbyintroducingspinintheproblem,wecanobtainaninsulatorthatmaintainsTRsymmetrybuthasarobust,gaplesspairofhelical(notchiral)edgestates.InthischapterweintroducetheKaneandMelemodelfirstandthenintroducetheHgTemodelforatopologicalinsulator.Suchaninsulatorwasthefirstexperimentallyrealizabletopologicalinsulator.TheKaneandMeleModelSOC:Forspin↑,theHamiltoniansatKandK′areThisisHaldanemodel.TheHaldanemasstermis

Lamda_so.

Assuch,inthisregime,fromouranalysisoftheHaldanemodelinthepreviouschapter,weknowthatthespin↑hasaHallconductanceequalto1.Forspin↓,theHamiltoniansatKandK′areAssuch,fromouranalysisoftheHaldanemodel,weknowthatthespin↑hasaHallconductanceequalto?1.Aswehavebothchiralandantichiralmodesonthesameedge(i.e.,statestravelinginbothdirectionsincloseproximity),agapwouldusuallyopenduetobackscattering.However,inthiscase,backscatteringsingle-particletermsareforbiddenduetoTRinvariance.Weprovedinchapter4thatthescatteringmatrixelementsbetweenTR-invariantpairsarezeroforanoddfermionnumber.Thismeansthatifwehaveanoddnumberoffermionpairsofedgemodes(oddnumberofKramers’doublets),wecannotopenagapbyasingle-particlebackscatteringterm—nosuchTR-invarianttermscanbewritten(howeverTR-invariantmultiparticleinteractiontermscanbewritten).IfwehaveanevennumberofKramers’pairsonanedge,wecanwritesingle-particlebackscatteringterms.Thissuggeststheexistenceoftwoseparateclassesand,thus,aZ2

classificationofnoninteractingtopologicalinsulators.MasstermBHZmodel

HgTe-CdTeQuantumWellsBHZmodel

HgTe-CdTeQuantumWellsSixbandsasix-componentspinorinthe3Dbulk:Inquantumwellsgrowninthe[001]direction,thecubic,orsphericalsymmetry,isbrokendowntotheaxialrotationsymmetryintheplane.Thesesix

bandscombinetoformthespin↑andspin-↓(±)statesofthreequantumwellsubbandsthat

havebeenlabeledasE1,H1,L1.TheL1subbandisseparatedfromtheothertwo,andweneglectit,leavinganeffectivefour-bandmodelforthinquantumwells.Fourbands-effectivemodelAtthisForthicknessd<dc,i.e.,fora

thinHgTelayer,thequantumwellisinthe“normal"regime,wheretheCdTeispredominant.“inverted"regimeExperimentalDetectionoftheQuantumSpinHallStateQHchiralstateZ2insulatoredgestatesEvencrossoddcrossTopologicalinvariantII:Z2Z2topologicalinvariantinT-symmetryinvariantsystem

Inthefollowingtwocases,Z2canbecalculatedeasilyCase1:SzisconservedoperatorCase2:Withinversionsymmetry,parityisagoodnumberQSHEinSilicene,Germanene,andStaneneExperimentalprogressesinSilicene,Germanene,andStaneneSiliceneSideViewTopViewBrillouinzoneLatticeConstant:3.86?;Bondlength:2.28?<2.35?inbulkSiliconΘ=101.73°EpitaxialgrowthofasilicenesheetSubstrateReferencesSiliceneAg(111)Vogtetal.,Phys.Rev.Lett.108,155501(2012)Chenetal.,Phys.Rev.Lett.109,056804(2012)Linetal.,Appl.Phys.Exp.5,045802(2012)Ir(111)Mengetal.,NanoLett.5,045802(2013)Au(110)Tchalalaetal.,Appl.Phys.Lett.102,083107(2013)ZrB2(0001)

Fleurenceetal.,Phys.Rev.Lett.108,245501(2012)Si-superlatticeAg-superlatticeExperimentalLatticeConstants3×34×41.14nma,d

1.15nmb1.18nmc√7×√7√13×√131.04nmb

1.08nmd√7×√72√3×2√31.0nmc

1.0nmd

0.87nmf,g

√3×√3?0.64nmc0.64±0.01nme

Ref.aVogt,P.etal.PhysicalReviewLetters108,155501(2012).Ref.bLin,C.-L.etal.AppliedPhysicsExpress5,045802(2012).Ref.cFeng,B.etal.NanoLetters12,3507(2012).Ref.dJamgotchian,H.etal.JPhysCondensMatter24,172001(2012).Ref.eChen,L.etal.PhysicalReviewLetters109,056804(2012).Ref.fLalmi,B.etal.AppliedPhysicsLetters97,223109(2010).Ref.gEnriquez,Hetal.JPhysCondensMatter24,314211(2012).

ThereconstructionofSiliceneonAg(111)surfaceAllsilicenephasesdependonthespecificgrowthconditionsincludinggrowthtemperature,coverage,substrate,etc.Togrow(1x1)siliceneoninsulatorsurfaceisstillchallenging!Silicenefield-effecttransistorsoperatingatroomtemperatureThree-dimensionalrenderingofAFMimageonasiliceneFETdeviceIdversusVgcurveofsiliceneFETdevicedisplaysambipolarelectron–holesymmetryexpectedfromsiliceneatroom-temperatureL.Taoetal.,10.1038/nnano.2014.325EpitaxialgrowthofaGermaneneonMetal(111)√3X√3Germaneneon√7X√7Au(111)surfaceM.E.Davilaetal.,NJP16(2014)095002ContinuousGermaneneLayeronAl(111)M.Derivazetal.,NanoLett.2015,15,2510Adv.Mater.2014,26,4820L.Liet.al,3X3Germaneneon√19X√19Pt(111)surfaceEpitaxialgrowthoftwo-dimensionalstaneneBluedottedlines:stanene.Greendashedlines:Bi2Te3(111)ElectronicstructuresofstanenefilmAtomicstructuresofstaneneonBi2Te3:STMtopographyF.-f.Zhuetal.,Nat.Mater.10.1038/nmat4384,2015LargeareaHRTEMimageofhexagonalstanenelatticeS.Saxenaetal.,Arxiv1505.05062QSHEinSilicene,Germanene,andStaneneCheng-ChengLiu,WanXiangFengandYuguiYao,PRL107,076802(2011)Cheng-ChengLiu,HuaJiangandYuGuiYao,PRB84,195430(2011)SiliceneSideViewTopViewBrillouinzoneLatticeConstant:3.86?;Bondlength:2.28?<2.35?inbulkSiliconΘ=101.73°Kane-Mele’sQuantumSpinHalleffectKineticenergyIntrinsicSOCRashbaSOCRashba:couplingtothesubstrateortheexternalelectricfield.Intrinsic:ueV0.5mKPRL95,226801(2005)QSHE:firstproposalingrapheneisunrealsitic!Yaoetal.PhysRevB.75.041401(2007)Secondorderprocess

toosmallgapGap=2ξ~1μeVsocsocplanarEffectiveSOCstrength:HowtoincreaseeffectiveSOCGapIncreaseatomicSOCstrength:

Zisatomicnumber

forsiliconatomξ~0.1meV:planarsilicene

Structureselection:

low-buckledstructureOrbitselection:Px,PyorbitsInvolve1storderprocessTheadiabaticevolutionofthegapAdiabaticcontinuityofGapQSHEStateinplanarstructurePlanarSiliceneLowbuckledSiliceneGapnotclosed!ThetopologicalinvariantZ2oflow-buckledsiliceneZ2

“density”inthe2DBrillouinzoneBlackdot=1;Whitedot=-1;empty=0Z2numberequalsthesumofZ2densityinhalfoftheBZTotalZ2mod2=1(2DtopologicalinsulatororQSHE)Itshouldbenotedthatdifferentgaugechoicesresultindifferentn-fieldconfigurations(Z2“density”);however,Z2isgaugeinvariant.TheSOCGap＆FermiVelocityvs.HydrostaticStrain

Fermivelocity:106m/sinGrapheneQSHEcanbemoreeasilyrealizedinSiliceneundercompressivestrainLowbuckled2DGermaniumwithhoneycombstructureStillQSHEstatePlanarhoneycombStrcutureLow-buckledhoneycombStrcuture~277KLatticeConstant:4.02?;Bondlength:2.42?<2.45?inbulkGermanium;Θ=106.5°Lowbuckled2DTinwithhoneycombstructurePlanarhoneycombStrcutureLow-buckledhoneycombStrcuture~852KLatticeConsttant:4.70?;Bondlength:2.84?<2.81inα-Sn;Θ=107.1°StillQSHEstatephononSOCgapfromFPcalculationsPlanarstructureLow-buckledstructureGraphene0.0008Silicene0.071.552DGemanium4.023.92DTin32.373.5Unit:meVWhyisSOCgapsobigforsiliceneet.al?Whatismicroscopicmechanism?HowdoesSOCgapdependonθ?

Tight-bindingmodel：

SymmetryAnalyse(I)

AccordingtotheparallelcomponentofforceF

SOC:AccordingtotheperpendicularcomponentofforceF

ThistermiscalledintrinsicRashbaSOC,itwillvanishinplanarstructure(e.g.Graphene)!Tight-bindingmodel：SymmetryAnalyse(II)

Then,weintroduceasecondnearestneighbortightbindingmodelByperformingFouriertransformations,thelow-energyeffectiveHamiltonianaroundDiracpointKinthebasist,t1,t2

areundeterminatedparameters,especially,wedon’tknowhowtheydependsangleθ!Tight-bindingmodel：low-energyeffectiveHamiltonian(I)Basis:Parameters:Slater-KosterTight-bindingmodelwithoutSOCatKpointTight-bindingmodel：low-energyeffectiveHamiltonian(II)Basis:Slater-KosterTight-bindingmodelwithSOCatKpointBylongcalculations,wecanobtainlow-energyeffectiveHamiltonianinbasisFordetails,pleaseseePRB84,195430(2011)Tight-bindingmodel：low-energyeffectiveHamiltonian(III)Here,Thefirst-ordereffectiveSOCstrength:Thesecond-ordereffectiveSOCstrength:Whenθ=90°,forplanarstructure.EffectiveSOCTight-bindingmodel：low-energyeffectiveHamiltonian(III)TheintrinsicRashbaSOCstrength:Fermivelocity:Comparedwiththesecondnearestneighbortightbindingmodel,wecandeterminetheparameterst,t1,t2:Whenθ=90°,TheintrinsicRashbaSOCvanishesatDiracpoint,theSOCgapwillnotbeaffectedbyIR-SOC,althoughthereisfinitevaluewhenkdeviatingfromDiracpoint.Therefore,itisverydifferentfromtheextrinsicRSOC,whichcandestroytheQSHE.ThevariationofgapopenedbySOCatDiracpointwiththeangleforsilicenefromTBFirstorderprocess:Secondorderprocess:RedLine:1storderSOCgapDotline:2ndorderSOCgapBlackline:thetotalgapApplicationsofthelow-energyeffectiveHamiltonianSlater-KosterbondparametersConclusionsinSilicene,etc.Byfirst-principlescalculations,topredictthatQSHEinsilicenecanbeobservedinanexperimentallyaccessiblelowtemperatureregimewiththespinorbitbandgapof1.55meV.2DGermanium/Tinwithlowbuckledstructurehavesimilarcase.Toanalyticallyderivethelow-energyeffectiveHamiltonianforlow-buckledhoneycombstructure,whichisverygeneral.a)ExcepttheintrinsicRashbaSOCterm,thelow-energyeffectiveHamiltonianisverysimilartothatofgraphene.However,effectiveSOCmainlycomesfromthefirst-orderprocess,notthesecond-orderprocess,thuseffectiveSOClargelyincreases.b)TheintrinsicRashbaSOCvanishesatDiracpoint,notlikeextrinsicRashbaSOC,itdoesnotdestroyQSHEstateofSilicene.PRL107,076802(2011);PRB84,195430(2011)3DTIs

WeakandstrongTIsTime-reversal-invariantmomenta(TRIMs)Aunitarymatrixisdefinedas:ExperimentalDetectionofStrongTIsHsieh,D.,D.Qian,L.Wray,Y.Xia,Y.S.Hor,R.J.Cava,andM.Z.Hasan,2008,Nature,London,452,970.ProposalsforWeakTIsTopologicalinsulatorsin3D3DTIsStrongTIsν0≠0WeakTIsν0=0;atleastoneofνi(i=1,3)≠0.3Dtopologicalinsulatorshave4z2indices:

ν0;(ν1,ν2,ν3)StrongTIshaveoddnumbersurfacestates;WeakTishaveevennumbersurfacestates.L.Fu,etal.PRB,2007.J.Mooreetal.PRB2007.Anar