《凝聚態(tài)物理學(xué)新論》配套教學(xué)課件

凝聚態(tài)物理學(xué)講課內(nèi)容第0章

拓?fù)淠軒Ю碚摵喗榈谝徽?/p>

（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?？茖W(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(shí)成績（66％）＋Project（34％）1.平時(shí)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(xiàn)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?fù)淠軒Ю碚摵喗檎n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?fù)浣^緣體（強(qiáng)或弱拓?fù)洌?，高階拓?fù)浣^緣體，拓?fù)浒虢饘伲―irac或者Weyl）以及它們的模型和材料實(shí)現(xiàn)。同時(shí)講解基本的理論知識(shí)，如Berryphase，Chernnumber，windingnumber，Z2number等拓?fù)洳蛔兞浚约巴負(fù)淠軒Ю碚摰?。凝聚態(tài)物理學(xué)進(jìn)入拓?fù)?的新時(shí)代！第0章

拓?fù)淠軒Ю碚摵喗橥負(fù)淞孔佑?jì)算低功耗自旋電子學(xué)新型紅外光電探測拓?fù)湮锢硌芯康闹匾獞?yīng)用價(jià)值拓?fù)湮锢硌芯康闹卮罂茖W(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(jiǎng)第一章傳統(tǒng)的凝聚態(tài)理論重要概念:

對(duì)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(fèi)米液體理論和朗道對(duì)稱性破缺理論對(duì)稱性對(duì)稱性（symmetry）是現(xiàn)代物理學(xué)中的一個(gè)核心概念，系統(tǒng)從一個(gè)狀態(tài)變換到另一個(gè)狀態(tài)，如果這兩個(gè)狀態(tài)等價(jià)，則說系統(tǒng)對(duì)這一變換是對(duì)稱的?；蛘哒f給系統(tǒng)一個(gè)“操作”，如果系統(tǒng)從一個(gè)狀態(tài)變到另一個(gè)等價(jià)的狀態(tài)，則說系統(tǒng)對(duì)這一操作是對(duì)稱的。它泛指規(guī)范對(duì)稱性（gaugesymmetry，或局域?qū)ΨQ性localsymmetry）和整體對(duì)稱性（globalsymmetry）。它是指一個(gè)理論的拉格朗日量或運(yùn)動(dòng)方程在某些變量的變化下的不變性。如果這些變量隨時(shí)空變化，這個(gè)不變性被稱為規(guī)范對(duì)稱性，反之則被稱為整體對(duì)稱性。物理學(xué)中最簡單的對(duì)稱性例子是牛頓運(yùn)動(dòng)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(duì)稱性。對(duì)稱性的性質(zhì)對(duì)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(dòng)、分子的轉(zhuǎn)動(dòng)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(dòng)。對(duì)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(duì)稱性。對(duì)稱性分類自然界的四類對(duì)稱性:(1)全同粒子的互換(2)連續(xù)時(shí)空變換,如平移,旋轉(zhuǎn)和加速

(3)分立變換,如空間反演,時(shí)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(duì)稱

對(duì)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(gè)不可觀測量變成可觀測的,對(duì)稱性就破缺了.對(duì)稱性與守恒定律物理系統(tǒng)的每一個(gè)對(duì)稱性都有相對(duì)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(duì)稱性。例如，空間位移對(duì)稱造成動(dòng)量守恒。而時(shí)間平移對(duì)稱造成能量守恒：為何過去和現(xiàn)在事物運(yùn)動(dòng)的規(guī)律是相同的？那是因?yàn)檫\(yùn)動(dòng)規(guī)律在時(shí)間平移的變動(dòng)中能夠保持不變。對(duì)稱破缺對(duì)稱性破缺（symmetrybreaking）系指物理學(xué)里，在具有某種對(duì)稱性的物理系統(tǒng)之臨界點(diǎn)附近發(fā)生的微小振蕩，通過選擇所有可能分岔中的一個(gè)分岔，打破了這物理系統(tǒng)的對(duì)稱性，并且決定了這物理系統(tǒng)的命運(yùn)。

自發(fā)對(duì)稱性破缺自發(fā)對(duì)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(duì)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(duì)稱性。通常，這種對(duì)稱性破缺會(huì)具有一種有序參數(shù)。自發(fā)對(duì)稱性破缺對(duì)稱破缺與Goldstone定理與對(duì)稱性破缺相關(guān)的一個(gè)結(jié)論是Goldstone定理:它是指在具有連續(xù)對(duì)稱性破缺的相對(duì)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(tuán)激發(fā)的聲子。晶體只有離散的平移對(duì)稱性，破缺了連續(xù)的平移對(duì)稱性。周光召先生:“對(duì)稱性和對(duì)稱破缺是世界統(tǒng)一性和多樣性的根源”生命起源中的對(duì)稱破缺：DNA左右鏡像對(duì)稱破缺！A-DNA、B-DNAZ-DNA右手雙螺旋左手雙螺旋藝術(shù)中的對(duì)稱性破缺維納斯女神《向日葵》梵高1888（荷蘭）自然界中的對(duì)稱破缺

凝聚態(tài)中的對(duì)稱性破缺凝聚態(tài)物質(zhì)世界大多數(shù)是對(duì)稱破缺的產(chǎn)物:晶體是平移對(duì)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(duì)稱性的破缺產(chǎn)生了鐵電體;時(shí)間反演對(duì)稱性的破缺產(chǎn)生磁有序結(jié)構(gòu);規(guī)范對(duì)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(gè)標(biāo)志,描述偏離對(duì)稱的性質(zhì)和程度.為某個(gè)物理量的平均值,可以是標(biāo)量,矢量,復(fù)數(shù)或更加復(fù)雜的量，是一個(gè)局域的量.隨對(duì)稱性的不同,它在高溫時(shí)為零,而低溫下取有限值,在Tc處轉(zhuǎn)變.對(duì)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(gè)多粒子系統(tǒng)在不同的溫度和壓強(qiáng)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(biāo)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(qiáng)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(jí)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(jí)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(jí)相變理論Landau的二級(jí)相變理論:

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

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

：矢量，在相變點(diǎn)，將自由能展開：不含奇次冪項(xiàng)要求：高于相變溫度時(shí)，

＝0使系統(tǒng)自由能達(dá)到極??；低于相變溫度時(shí)，

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

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

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

F(4)λ點(diǎn)均為溫度的緩變函數(shù),(不參與求導(dǎo))比熱在相變溫度點(diǎn)不連續(xù)：傳統(tǒng)凝聚態(tài)物理中的對(duì)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(qiáng)關(guān)聯(lián)下，電子系統(tǒng)被稱為電子液體或費(fèi)米液體或Luttinger液體(1D)

費(fèi)米溫度：均勻的無相互作用的三維系統(tǒng)，費(fèi)米溫度：費(fèi)米簡并系統(tǒng)：費(fèi)米子系統(tǒng)的溫度通常遠(yuǎn)低于費(fèi)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(fèi)米液體理論：

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

表示朗道費(fèi)米液體理論的適用條件：(1).必須有可明確定義的費(fèi)米面存在(2).準(zhǔn)粒子有足夠長的壽命朗道費(fèi)米液體理論是處理相互作用費(fèi)米子體系的唯象理論。在相互作用不是很強(qiáng)時(shí)，理論對(duì)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(fèi)米液體行為：與費(fèi)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(shí)包含了許多對(duì)稱性相同而本質(zhì)又不同的態(tài)。所以這些相就不能用對(duì)稱性加以區(qū)分，故也不能用朗道對(duì)稱破缺理論描述。這時(shí)，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?fù)涞母拍?。如symmetryprotectedphases（SPT），包括拓?fù)浣^緣體，拓?fù)涑瑢?dǎo)體，拓?fù)浒虢饘俚?。電子在電磁場中的運(yùn)動(dòng)（經(jīng)典）運(yùn)動(dòng)方程等式左邊第一項(xiàng)是加速度項(xiàng)，第二項(xiàng)是碰撞項(xiàng)；右邊是電子受到的Lorenz力。當(dāng)磁場B平行于z軸時(shí)，上述運(yùn)動(dòng)方程如下：對(duì)于靜電場中的穩(wěn)態(tài)，時(shí)間導(dǎo)數(shù)為0，于是漂移速度為回旋共振頻率。霍爾效應(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(shí)將會(huì)產(chǎn)生橫跨導(dǎo)體兩個(gè)面的電場，其方向?yàn)閖xB，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(duì)于電子，取負(fù)號(hào)。可用于載流子濃度測量，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(dá)式：其中的系數(shù)即為靜態(tài)磁致電導(dǎo)率張量。在強(qiáng)磁場下，wc*tau>>1，Hall電導(dǎo)率如下此時(shí)，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(qiáng)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(tái)。1.thevonKlitzingconstantRK=h/e2=25812.807557(18)Ω.被用作電阻標(biāo)準(zhǔn)。2.用于測量精細(xì)結(jié)構(gòu)常數(shù)e2/hc.3.要解釋為什么會(huì)有如此高

精度的量子化電導(dǎo)平臺(tái)（即使

在有一些雜質(zhì)的情況下），這就涉及了本

課程的重點(diǎn)：幾何相位，拓?fù)淠軒Ю碚摰?/p>

知識(shí)。播放動(dòng)畫Geometricangleandparalleltransportvneverrotatesaboute3paralleltransport:Wavefunctionparalleltransport:Gaugecovariantderivative:Geometricangle和傅科擺Paralleltransportofavectoraroundaclosedloop(fromAtoNtoBandbacktoA)onthesphere.Theanglebywhichittwists,\alpha,isproportionaltotheareainsidetheloop.傅科擺傅科擺擺動(dòng)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(shí)間，用小時(shí)作單位θ°=15°*t*sinφ播放動(dòng)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(rèn)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(rèn)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?fù)浣^緣體；拓?fù)浒虢饘佟@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(xiàn)量子反?；魻栃?yīng)的實(shí)驗(yàn)實(shí)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