【正文】 2. 窄能帶現(xiàn)象的理論模型選擇經(jīng)驗(yàn)參數(shù)的模型 Hamilton量方法Hubbard模型和 Anderson模型The Hubbard ModelFrom simple quantum mechanics to manyparticle interaction in solidsa short introductionHistorical facts? Hubbard Model was first introduced by John Hubbard in 1963.? Who was Hubbard? He was born in 1931 and died 1980. Theoretician in solid state physics, field of work: Electron correlation in electron gas and small band systems. He worked at the ., Harwell, ., and at the IBM Research Labs, San Jos233。稀土化合物部分存在混價(jià) “mixed valence”。 (c). 一些過渡金屬氧化物當(dāng)溫度升高時(shí)會(huì)從絕緣體 ?金屬f電子或 d電子波函數(shù)的分布范圍是否和近鄰產(chǎn)生重疊，是電子離域還是局域化的基本判據(jù)l殼層體積與 WingerSeitz 元胞體積的比值：4f最小， 5f次之， 3d,4d,5d…多電子態(tài)的局域化強(qiáng)度的順序：4f5f3d4d5d______________能帶寬度上升另外，從左往右穿過周期表，部分填充殼層的半徑逐步降低，關(guān)聯(lián)重要性增加。s golden rule end to endWeak linkx = 0However, now is tunneling from the end of a LLCharge density wave is pinned at the impurityPHYSICAL REALIZATIONS Semiconducting quantum wiresEdge states in fractional quantum Hall effectSinglewalled metallic carbon nanotubesEFEnergymetallic 1D conductor with 2 linear bandsk 強(qiáng)關(guān)聯(lián)體系1. 窄能帶現(xiàn)象 金屬與絕緣體之分： (1)能帶框架下的區(qū)分：導(dǎo)帶 導(dǎo)帶價(jià)帶 價(jià)帶(2)無(wú)序引起的 Anderson 轉(zhuǎn)變：局域態(tài) 擴(kuò)展態(tài) 局域態(tài)局域態(tài)局域態(tài) 擴(kuò)展態(tài)EF EF(3)電子間關(guān)聯(lián)導(dǎo)致的 Mott金屬－絕緣體轉(zhuǎn)變 (a).MnO:5個(gè) 3d?未滿 3d帶； O2 2p是滿帶不與 3d能帶重疊 能帶論 ?MnO的 3d帶將具有金屬導(dǎo)電性 實(shí)際上， MnO是絕緣體！ (b).ReO3:能帶論 ?絕緣體。 yieldsWhat about the lifetime t in 1D?formally, it divergesat small qbut we can insert asmall cutoffAt small T., this ratio cannot bemade arbitrarily smallas in 3DBREAKDOWN OF LANDAU THEORY IN 1D1 2 3 40132DISPERSION OF EXCITATIONS IN 1D collective excitations are plasmons with (RPA)single particlegaplessplasmon COLLECTIVE AND SINGLEPARTICLE EXCITATION NON DISTINCT no longer diverges at (no angular integration over direction of as in 3D ) THE TOMONAGALUTTINGER MODELEXACTLY SOLVABLE MODEL FOR INTERACTING 1D ELECTRONS AT LOW ENERGIESDispersion relation is linearized near(both collective and singleparticle excitations have linear dispersion) Model bees exact when linearized branches extend from Assumptions:Only small momenta exchanges are includedTOMONAGALUTTINGER HAMILTONIANFree part free part interaction fermionic annihilation/creation operatorsIntroduce right moving k 0, and left moving k 0 electrons TL HAMILTONIAN IIInteractions free part interactionbackscattering forward umklapp forwardBOSONIZATIONBOSONIZATION: EXPRESS FERMIONIC HAMILTONIANIN TERMS OF BOSONIC OPERATORSconstruct bosonic Hamiltonian with the same spectrun(a) (b) (c) (d)(a) and (b) havesame spectrum butdifferent groundstateEXCITED STATE CAN BE WRITTEN IN TERMS OF CHARGEEXCITATIONS, OR BOSONIC ELECTRONHOLE EXCITATIONSSTEP 1WHICH OPERATORS DO THE JOB?Introduce the density operators (create excitation of momentum q)and consider their mutation relations nearly bosonic mutation relations STEP 1: PROOFConsider .algebra offermionic operatorsoccupation operatorSTEP 2Examine nowBOSONIZED HAMILTONIANSTATES CREATED BY ARE EIGENSTATES OF WITH ENERGY andinteractionsSTEP 2: PROOFExample:STEP 3Introduce the bosonic operatorsyieldingDIAGONALIZATIONSPINCHARGE SEPARATIONand interaction (satisfying SU2 symmetry)If we include spin, it gets slightly more plicated ... and interesting Introduce the spin and charge densitiesHamiltonian decouple in two independent spin and charge parts,with excitations propagating with velocities SPACE REPRESENTATIONLong wavelength limit (interactions )Appropriate linear binations P, q of the field ?(x) can be defined.Then one finds whereLuttinger parameter g 1 repulsive interactionBOSONIC REPRESENTATION OF YFermionic operatorWhere . Express y in the form of a bosonic displacement operator B ? from ? decreases the number of electrons by one? displaces the boson configuration for that stateBOSONIZATION IDENTITYif a cnumber U ladder operator, q bosonicLOCAL DENSITY OF STATESi) Local density of states at x = 0n density of states of noninteracting systemat T = 0ii) Local density of states at the end of a Luttinger liquid at T = 0cutoff energyG gamma functionMEASURING THE LDOS Measurement of the local density of states system 1system 2 couplingIVby tunnelingSee . carbon nanotube experiment by Bockrath et al. Nature, 397, 598 (1999)MEASURING THE LDOS IItunneling rate i to j Tunneling current can be evaluated by use of Fermi180。 IIIn 1D k, k180。s golden rule yields for the lifetime tT = 0LIFETIME OF ``QUASIPARTICLES180。180。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)THE PHYSICS OF LUTTINGER LIQUIDSFERMI SURFACE HAS ONLY TWO POINTSfailure of Landau180。一維情況下，不成立。在相互作用不是很強(qiáng)時(shí)，理論對(duì)三維液體正確。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。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)相互作用 ： (1)單電子能級(jí)分布變化 (勢(shì)的變化 )。2. 費(fèi)米液體 金屬中電子通常是可遷移的，稱為電子氣， 電子動(dòng)能：電子勢(shì)能：在高密度下，電子動(dòng)能為主，自由電子氣模型是較好的近似。）有限差分? 從微分到差分? 提高 FD方法的計(jì)算效率– 對(duì)網(wǎng)格進(jìn)行優(yōu)化，如曲線網(wǎng)格（適應(yīng)網(wǎng)格）和局部網(wǎng)格優(yōu)化（復(fù)合網(wǎng)格）– 結(jié)合贗勢(shì)方法 – 多尺度（ multiscale）或預(yù)處理（ preconditioning）有限元? 變分方法 ? 處理復(fù)雜的邊界條件? 矩陣稀疏程度及帶狀結(jié)構(gòu)往往不如有限差分好 ? 廣義的本征值問題多分辨網(wǎng)格上的小波基組? 多分辨分析? 半取樣（ semicardinal）基組線性標(biāo)度與量子力學(xué)中的局域性? “近視原理 ” ? 局域化的 Wannier函數(shù)或密度矩陣– 絕緣體：指數(shù)衰減，能隙越大衰減越快– 金屬：零溫下按冪率衰減，在有限溫度下可出現(xiàn)指數(shù)衰減? 局域區(qū)域 ? 線性標(biāo)度系數(shù)， crossover線性標(biāo)度算法? 分治方法 ? 費(fèi)米算符展開和費(fèi)米算符投影方法? 直接最小化方法– 密度矩陣最小化– 軌道最小化– 優(yōu)基組密度矩陣最小化線性標(biāo)度算法? 基于格林函數(shù)的遞歸方法 ? 脫離軌道的（ orbitalfree,OF）算法 ? 對(duì)角化以外的線性標(biāo)度– 構(gòu)造有效哈密頓量的算法 – 幾何優(yōu)化與分子動(dòng)力學(xué)– TDDFTPart III:應(yīng)用物理學(xué)：強(qiáng)相關(guān)體系? 模型哈密頓量 ? LDA++ ? 電子結(jié)構(gòu)： CrO2? 點(diǎn)陣動(dòng)力學(xué) : 钚化學(xué)：弱作用體系? 松散堆積的軟物質(zhì)、惰性氣體、生物分子和聚合物，物理吸附、 Cl+HD反應(yīng)? 用傳統(tǒng)的密度泛函理論處理弱作用體系? 一個(gè)既能產(chǎn)生 vdW相互作用系數(shù)又能產(chǎn)生總關(guān)聯(lián)能的非局域泛函：無(wú)縫的（seamless）方法? GW近似 ? 密度泛函加衰減色散（ DFdD）生命科學(xué)：生物體系? 困難（尺寸問題、時(shí)間尺度） ? QM/MM方法（飽和原子法、凍結(jié)軌道法）