【正文】 finite lifetime t3DFERMI LIQUIDS IIIcollectiveexcitations(plasmons)singleparticleexcitations1 2 3 40132DISPERSION OF EXCITATIONS IN 3D 0nointeractingT = 0Finite jump in momentum distributionZZ quasiparticle weightLIFETIME OF ``QUASIPARTICLES180。180。scattering out of state k scattering into state kspinscreened Coulomb interactionenergy conservationIn 3D an integration over angular dependence takes care of dfunction Fermi180。s golden rule yields for the lifetime tT = 0LIFETIME OF ``QUASIPARTICLES180。180。 IIIn 1D k, k180。 are scalars. Integration over k180。 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。s golden ruleconstant LL to LLLL to metalSINGLE IMPURITY Again tunneling current can be evaluated by use of Fermi180。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)帶價帶 價帶(2)無序引起的 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 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。實(shí)際上是金屬。 (c). 一些過渡金屬氧化物當(dāng)溫度升高時會從絕緣體 ?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。4f， 5f元素和 3d,4d,5d元素的殼層體積與 WingerSeitz元胞體積的比值YScSmith和 Kmetko準(zhǔn)周期表窄帶區(qū)域重費(fèi)米子強(qiáng)鐵磁性超導(dǎo)體離域性局域性另一類窄帶現(xiàn)象：來自能帶中的近自由電子與溶在晶格中具有 3d,5f或 4f殼層電子的溶質(zhì)原子相互作用 Friedel與 Anderson稀土元素或過渡金屬化合物中的能隙不可能僅用 “電荷轉(zhuǎn)移能 ”、 “雜化能隙 ”、 “有效庫侖相關(guān)能 ”三者之一來描述，而應(yīng)該說三者同時發(fā)揮作用。稀土化合物部分存在混價 “mixed valence”?；靸r的作用導(dǎo)致在 Fermi面附近存在非常窄的能帶 (部分填充 f能帶或 f能級），電子可以在 4f能級和離域化能帶之間轉(zhuǎn)移，對固體基態(tài)性質(zhì)產(chǎn)生顯著影響。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。, USA. Picture taken from: Physics Today, Vol. 34, No4, 1981What, in general, is the HM? ? Hubbard model is a quantum theoretical model for manyparticle interaction in and