第五章第五章 关联关联5.1 5.1 单电子近似的理论基础单电子近似的理论基础 5.2 5.2 费米液体理论费米液体理论5.3 5.3 强关联体系强关联体系多电子体系多电子体系(After Born-Oppenheimer (After Born-Oppenheimer 绝热近似绝热近似):):5.1 5.1 单电子近似的理论基础单电子近似的理论基础关联：电子电子相互作用关联：电子电子相互作用弱：单电子近似，弱：单电子近似， 电子平均场电子平均场1. Hartree方程方程(1928)连乘积形式：连乘积形式：按变分原理，按变分原理， 的选取的选取E达到极小达到极小正交归一条件正交归一条件单电子方程单电子方程动能动能原子核对电子形成的势能原子核对电子形成的势能其余其余N-1N-1个电子对个电子对j j电子的库仑作用能电子的库仑作用能自洽求自洽求解，解，H H2 2,He,He计算与实验相符。计算与实验相符。2626个电子的个电子的FeFe原子，运算要涉及原子，运算要涉及10107676个数，对称简化个数，对称简化10105353个个整个太阳系没有足够物质打印这个数据表！整个太阳系没有足够物质打印这个数据表！2. 凝胶模型凝胶模型(jellium model)为突出探讨相互作用电子系统的哪些特征是区别于不计其相为突出探讨相互作用电子系统的哪些特征是区别于不计其相互作用者，可人为地简化假定电子是沉浸在空间密度持恒的互作用者，可人为地简化假定电子是沉浸在空间密度持恒的正电荷背景之中正电荷背景之中(不考虑离子的周期性不考虑离子的周期性)。正电荷的作用体现于在相互作用电子体系的正电荷的作用体现于在相互作用电子体系的Hamiltonian中出中出现一个维持系统聚集的附加项现一个维持系统聚集的附加项金属体系，设电子波函数：金属体系，设电子波函数：HartreeHartree方程中的势：方程中的势：第二项是全部电子在第二项是全部电子在r r处形成的势，与处形成的势，与 相抵消相抵消第三项是须扣除的自作用，与第三项是须扣除的自作用，与j j有关，但如取有关，但如取r r为计算原点：为计算原点：所以对凝胶模型，所以对凝胶模型，HartreeHartree方程：方程：相互作用相互作用没有相互作用没有相互作用 电子正电荷背景电子正电荷背景自由自由电子气子气3. Hartree-Fock方程方程(1930) Hartree方程不满足方程不满足Pauli不相容原理不相容原理 电子：费米子电子：费米子 单电子波函数单电子波函数f f：N N电子电子 体系的体系的总波函数波函数： 不涉及自旋轨道耦合时：不涉及自旋轨道耦合时：N N电子体系能量期待值：电子体系能量期待值：1.1.第二项第二项j,jj,j可可以相等，自相互以相等，自相互作用作用2.2.自相互作用严自相互作用严格相消（通过第格相消（通过第二，三项）二，三项）3.3.第三项为交换第三项为交换项，同自旋电子项，同自旋电子通过变分通过变分: :么正变换么正变换: :单电子方程：单电子方程：与与HartreeHartree方程的差别：第三项对全体电子，第四项新增，交方程的差别：第三项对全体电子，第四项新增，交换作用项。求和只涉及与换作用项。求和只涉及与j j态自旋平行的态自旋平行的jj态，是电子服从态，是电子服从FermiFermi统计的反映。统计的反映。4. Koopmann定理（定理（1934)单电子轨道能量等于单电子轨道能量等于N N电子体系从第电子体系从第j j个轨道上取走一个电子个轨道上取走一个电子并保持并保持N N1 1个电子状态不不变的总能变化值。个电子状态不不变的总能变化值。推广：系统中一个电子由状态推广：系统中一个电子由状态j j转移到态转移到态i而引起系统能量的变化而引起系统能量的变化5. 交换空穴交换空穴(Fermi hole) 将将H-F方程改写为：方程改写为：其中：其中：定性讨论：假设定性讨论：假设Fermi hole:与某电子自与某电子自旋相同的其余邻近电子旋相同的其余邻近电子在围绕该电子形成总量在围绕该电子形成总量为为1的密度亏欠域的密度亏欠域energy as a function of the one electron density, nuclear-electron attraction, electron-electron repulsionThomas-Fermi approximation for the kinetic energySlater approximation for the exchange energy 6. 密度泛函理论密度泛函理论(Density functional theory) (1) Thomas-Fermi-Dirac Model(2) The Hohenberg-Kohn Theorem properties are uniquely determined by the ground-state electron In 1964, Hohenberg and Kohn proved thatmolecular energy, wave function and all other molecular electronic probability density namely,Phys. Rev. 136, 13864 (1964) .”Density functional theory (DFT) attempts toand other ground-state molecular properties from the ground-state electron density “For molecules with a nondegenerate ground state, the ground-state calculate Proof:The electronic Hamiltonian isit is produced by charges external to the system of electrons.In DFT, is called the external potential acting on electron i, sinceOnce the external potential the electronic wave functions and allowed energies of the molecule are and the number of electrons n are specified, determined as the solutions of the electronic Schrdinger equation. Now we need to prove that the ground-state electron probability density the number of electrons. the external potential (except for an arbitrary additive constant) a) Sincedetermines the number of electrons.b) To see thatdetermines the external potential, we supposethat this is false and that there are two external potentialsand(differingby more than a constant) that each give rise to the same ground-state electrondensity.determinesthe exact ground-state wave function and energy of the exact ground-state wave function and energy of LetSinceanddiffer by more than a constant,andmust be different functions.Proof:Assume thusthuswhich contradicts the giveninformation.function, the exact ground-state wave function state energy for a given Hamiltonian If the ground state is nondegenerate, then there is only one normalizedthat gives the exact groundAccording to the variation theorem, suppose that If thenis any normalizedwell-behaved trial variation function. Now use as a trial function with the HamiltonianthenSubstituting givesLetbe a function of the spatial coordinatesof electron i,thenUsing the above result, we getSimilarly, if we go through the same reasoning with a and b interchanged, we getBy hypothesis, the two different wave functions give the same electron. Putting and adding the above two inequalitiesdensity: yieldpotentials could produce the same ground-state electron density must be false. energy) and also determines the number of electrons. This result is false, so our initial assumption that two different externalpotential (to within an additive constant that simply affects the zero level ofHence, the ground-state electron probability density determines the externalprobability densityand other properties”emphasizes the dependence of the external potential differs for different molecules.“For systems with a nondegenerate ground state, the ground-state electrondetermines the ground-state wave function and energy, whichHowever, the functionalsare unknown.is also written asThe functionalindependent of the externalonispotential.(3) The Hohenberg-kohn variational theorem“For every trial density functionthat satisfiesandfor all, the following inequality holds:, is the true groundstate energy.”Proof:Letsatisfy thatandHohenberg-Kohn theorem, determines the external potential and this in turn determines the wave functiondensity . By the,that corresponds to the .wherewith Hamiltonian. According to the variation theoremLet us use the wave functionas a trial variation function for the moleculeSince the left hand side of this inequality can be rewritten asOne gets states. Subsequently, Levy proved the theorems for degenerate ground states. Hohenberg and Kohn proved their theorems only for nondegenerate ground(4) The Kohn-Sham method If we know the ground-state electron density molecular properties fromfunction., the Hohenberg-Kohntheorem tells us that it is possible in principle to calculate all the ground-state, without having to find the molecular wave 1965, Kohn and Sham devised a practical method for finding andfor finding from. Phys. Rev., 140, A 1133 (1965). Their method is capable, in principle, of yielding exact results, but because the equations of the Kohn-Sham (KS) method contain an unknown functional that must beapproximated, the KS formation of DFT yield approximate results.沈吕九沈吕九electrons that each experience the same external potential the ground-state electron probability density equal to the exact of the molecule we are interested in:. Kohn and Sham considered a fictitious reference system s of n noninteractingthat makesof the reference systemSince the electrons do not interact with one another in the reference system,the Hamiltonian of the reference system iswhereis the one-electron Kohn-Sham Hamiltonian. Thus, the ground-state wave functionof the reference system is: is a spin functionorbital energies.are Kohn-ShamFor convenience, the zero subscript on is omitted hereafter.Defineas follows:ground-state electronic kinetic energysystem of noninteracting electrons.(either)is the difference in the averagebetween the molecule and the reference The quantityrepulsion energy.units) for the electrostatic interelectronic is the classical expression (in atomicRemember thatWith the above definitions, can be written asDefine the exchange-correlation energy functional byNow we haveside are easy to evaluate fromget a good approximation to to the ground-state energy. The fourth quantity accurately. The key to accurate KS DFT calculation of molecular properties is to The first three terms on the rightis a relativelysmall term, but is not easy to evaluate and they make the main contributionsThusbecomes.Now we need explicit equations to find the ground-state electron density.same electron density as that in the ground state of the molecule: is readily proved thatSince the fictitious system of noninteracting electrons is defined to have the, itground-state energy by varying to minimize the functional can vary the KS orbitals minimize the above energy expression subject to the orthonormality constraint: The Hohenberg-Kohn variational theorem tell us that we can find the so as. Equivalently, instead of varyingweThus, the Kohn-Sham orbitals are those thatwith the exchange-correlation potential defined by(If is known, its functional derivative is also known.)Comments on the DFT methods:(1) The KS equations are solved in a self-consistent fashion, like the HF equations.(2) The computation time required for a DFT calculation formally scales the third power of the number of basis functions.(3) There is no DF molecular wave function.(4) The KS orbitals can be used in qualitative MO discussions, like the HF orbitals.(5) Koopmans theorem doesnt hold here, exceptThe KS operator exchange operators in the HF operator are replaced by the effects of both exchange and electron correlation. is the same as the HF operator except that the, which handles(6) Various approximate functionals DF calculations. The functional and a correlation-energy functionalAmong variousCommonly used andPW91 (Perdew and Wangs 1991 functional)Lee-Yang-Parr (LYP) functionalare used in molecularapproximations, gradient-corrected exchange andcorrelation energy functionals are the most accurate.PW86 (Perdew and Wangs 1986 functional)B88 (Beckes 1988 functional)P86 (the Perdew 1986 correlation functional) (7) Nowadays KS DFT methods are generally believed to be better thanthe HF method, and in most cases they are even better than MP2 is written as the sum of anexchange-energy functional X Local exchangeApproximate density functional theories for exchange and correlationX : Local exchange functional of the homogeneous electron gasLDALocal exchange +local correlationGGALocal exchange +local correlation +gradient corrections3rd Generation of functionalsLDA: Local exchange functional + local correlation functional of the homogeneous electron gasGGA: Same as LDA + “non-local” gradient corrections to exchange and correlation3rd Generation of functionals: Same as GGA + instilation of “exact-exchange” and + 2nd derivativesof the density correctionsTerms in Density FunctionalsLocal densityrsSeitz radius = (3/4p)1/3kFFermi wave number = (3p2)1/3tDensity gradient = |grad |/2fkszSpin polarization = (up - down)/fSpin scaling factor = (1+z)2/3 + (1-z)2/3/2ksThomas-Fermi screening wave number = (4kF/pa0)1/2sAnother density gradient = |grad |/2kFJ.Chem.Phys. ,100,1290(1994);PRL77,3865(1996).Local Density ApproximationLocal Spin Density ApproximationLocal Spin Density Correlation FunctionalNot for the faint of heart:Generalized Gradient Approximation FunctionalsThe Nobel Prize in Chemistry 1998“for his development of the density-functional theory Walter Kohn (1923-)5.2 5.2 费米液体理论费米液体理论1.费米体系费米体系 费米温度：费米温度：均匀的无相互作用的三维系统，费米温度：均匀的无相互作用的三维系统，费米温度：费米简并系统：费米子系统的温度通常运运低于费米温度费米简并系统：费米子系统的温度通常运运低于费米温度 室温下金属中的传导电子室温下金属中的传导电子费米温度给出了系统中元激发存在与否的标度费米温度给出了系统中元激发存在与否的标度在费米温度以下，系统的性质由数目有限的低激发态决定。在费米温度以下，系统的性质由数目有限的低激发态决定。有相互作用和无相互作用的简并费米子系统中，低激发态的有相互作用和无相互作用的简并费米子系统中，低激发态的性质具有较强的对应性。性质具有较强的对应性。2. 费米液体费米液体 金属中电子通常是可迁移的，称为电子气，金属中电子通常是可迁移的，称为电子气， 电子动能：电子动能：电子势能：电子势能：在高密度下，电子动能为主，自由电子气模型是较好的近在高密度下，电子动能为主，自由电子气模型是较好的近似。在低密度下，电子之间的势能或关联变得越来越重要，似。在低密度下，电子之间的势能或关联变得越来越重要，电子可能由于这种关联作用进入液相甚至晶相。电子可能由于这种关联作用进入液相甚至晶相。较强关联下，电子系统被称为较强关联下，电子系统被称为电子液体电子液体或或费米液体费米液体或或Luttinger液体液体(1D)相互作用相互作用： (1)单电子能级分布变化单电子能级分布变化(势的变化势的变化);(2)电子散电子散射导致某一态上有限寿命射导致某一态上有限寿命(驰豫时间驰豫时间)3. 朗道费米液体理论朗道费米液体理论 单电子图象不是一个正确的出发点，但只要把电子改成准单电子图象不是一个正确的出发点，但只要把电子改成准粒子或准电子，就能描述费米液体。准粒子遵从费米统计，粒子或准电子，就能描述费米液体。准粒子遵从费米统计，准粒子数守恒，因而费米面包含的体积不发生变化。准粒子数守恒，因而费米面包含的体积不发生变化。假设激发态用动量假设激发态用动量 表示表示朗道费米液体理论的适用条件：朗道费米液体理论的适用条件：(1). 必须有可明确定义的费米面存在必须有可明确定义的费米面存在(2). 准粒子有足够长的寿命准粒子有足够长的寿命Fermi Liquid TheorySimple Picture for Fermi Liquid朗道费米液体理论是处理相互作用费米子体系的唯象理论。朗道费米液体理论是处理相互作用费米子体系的唯象理论。在相互作用不是很强时，理论对三维液体正确。在相互作用不是很强时，理论对三维液体正确。二维情况下，多大程度上成立不知道。二维情况下，多大程度上成立不知道。一维情况下，不成立。一维情况下，不成立。luttinger液体液体一维：低能激发为自旋为一维：低能激发为自旋为1/2的电中性自旋子和无自旋荷电为的电中性自旋子和无自旋荷电为 的波色子的激发。的波色子的激发。非费米液体行为：与费米液体理论预言相偏离的性质非费米液体行为：与费米液体理论预言相偏离的性质THE PHYSICS OF LUTTINGER LIQUIDSFERMI SURFACE HAS ONLY TWO POINTSfailure of Landaus Fermi liquid pictureELECTRONS FORM A HARMONIC CHAIN AT LOW ENERGIES Coulomb + Pauli interactionTHE LUTTINGER LIQUID: INTERACTING SYSTEM OF 1D ELECTRONS AT LOW ENERGIEScollective excitations are vibrational modesREMARKABLE PROPERTIESAbsence of electron-like quasi-particles(only collective bosonic excitations)Spin-charge separation(spin and charge are decoupled and propagate with different velocities)Absence of jump discontinuity in the momentum distribution at Power-law behavior of various correlation functions and transportquantities. The exponent depends on the electron-electron interactionOUTLINEWhat is a Fermi liquid, and why the Fermi liquid concept breaks in 1DThe Tomonaga-Luttinger model The TL-Hamiltonian and its bosonization Diagonalization Bosonic fields and electron operators Local density of states Tunneling into a Luttinger liquidLuttinger liquid with a single impurityPhysical realizations of Luttinger liquidsLITERATURE K. FlensbergLecture notes on the one-dimensional electron gas and the theory of Luttinger liquids J. von Delft and H. Schoeller Bosonization for beginners refermionization for experts, cond-mat/9805275J. VoitOne-dimensional Fermi liquids, Rep. Prog. Phys. 58, 977 (1995)H.J. Schulz, G. Cuniberti and P. PieriFermi liquids and Luttinger liquids, cond-mat/9807366SHORTLY ABOUT FERMI LIQUIDSLandau 1957-1959Also collective excitations occur (e.g. zero sound) at finite energiesLow energy excitations of a system of interacting particles described in terms of quasi-particles (single-particle excitations) Key point: quasi-particles have same quantum numbers as the corresponding non-interacting system (adiabatic continuity)Start from appropriate noninteracting systemRenormalization of a set of parameters (e.g. effective mass)FERMI LIQUIDS IIPauli exclusion principle only states within kT around Fermi sphere available quasiparticle states near Fermi sphere scatter only weaklyQUASI-PARTICLE PICTURE IS APPLICABLE IN 3D Effect of Coulomb interaction is to induce a finite life-time t3DFERMI LIQUIDS IIIcollectiveexcitations(plasmons)single-particleexcitations12340132DISPERSION OF EXCITATIONS IN 3D 0nointeractingT = 0Finite jump in momentum distributionZZ quasi-particle weightLIFETIME OF QUASI-PARTICLESscattering out of state kscattering into state kspinscreened Coulomb interactionenergy conservationIn 3D an integration over angular dependence takes care of d-function Fermis golden rule yields for the lifetime tT = 0LIFETIME OF QUASI-PARTICLES IIIn 1D k, k are scalars. Integration over k yieldsWhat about the lifetime t in 1D?formally, it divergesat small qbut we can insert asmall cut-offAt small Ti.e., this ratio cannot bemade arbitrarily smallas in 3DBREAKDOWN OF LANDAU THEORY IN 1D12340132DISPERSION OF EXCITATIONS IN 1D collective excitations are plasmons with (RPA)single particlegaplessplasmon COLLECTIVE AND SINGLE-PARTICLE EXCITATION NON DISTINCT no longer diverges at (no angular integration over direction of as in 3D ) THE TOMONAGA-LUTTINGER MODELEXACTLY SOLVABLE MODEL FOR INTERACTING 1D ELECTRONS AT LOW ENERGIESDispersion relation is linearized near(both collective and single-particle excitations have linear dispersion) Model becomes exact when linearized branches extend from Assumptions:Only small momenta exchanges are includedTOMONAGA-LUTTINGER HAMILTONIANFree part free partinteraction fermionic annihilation/creation operatorsIntroduce right moving k 0, and left moving k 0 electrons TL HAMILTONIAN IIInteractions free partinteractionbackscatteringforwardumklappforwardBOSONIZATIONBOSONIZATION: 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 ELECTRON-HOLE EXCITATIONSSTEP 1WHICH OPERATORS DO THE JOB?Introduce the density operators (create excitation of momentum q)and consider their commutation relations nearly bosonic commutation relations STEP 1: PROOFConsider e.g.algebra offermionic operatorsoccupation operatorSTEP 2Examine nowBOSONIZED HAMILTONIANSTATES CREATED BY ARE EIGENSTATES OF WITH ENERGY andinteractionsSTEP 2: PROOFExample:STEP 3Introduce the bosonic operatorsyieldingDIAGONALIZATIONSPIN-CHARGE SEPARATIONand interaction (satisfying SU2 symmetry)If we include spin, it gets slightly more complicated . 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 combinations P, q of the field (x) can be defined.Then one finds whereLuttinger parameter g 5f3d4d5d_能带宽度上升能带宽度上升另外，从左往右穿过周期表，部分填充壳层的半径逐步另外，从左往右穿过周期表，部分填充壳层的半径逐步降低，关联重要性增加。降低，关联重要性增加。4f，5f元素和元素和3d,4d,5d元素的壳层体积与元素的壳层体积与Winger-Seitz元胞体积的比值元胞体积的比值YScSmith和和Kmetko准周期表准周期表窄带区域窄带区域重费米子重费米子强铁磁性强铁磁性超导体超导体离域性离域性局域性局域性另一类窄带现象：来自能带中的近自由电子与溶在晶格中具另一类窄带现象：来自能带中的近自由电子与溶在晶格中具有有3d,5f或或4f壳层电子的溶质原子相互作用壳层电子的溶质原子相互作用 Friedel与与Anderson稀土元素或过渡金属化合物中的能隙不可能仅用稀土元素或过渡金属化合物中的能隙不可能仅用“电荷转移电荷转移能能”、“杂化能隙杂化能隙”、“有效库仑相关能有效库仑相关能”三者之一来描述，三者之一来描述，而应该说三者同时发挥作用。而应该说三者同时发挥作用。稀土化合物部分存在混价稀土化合物部分存在混价“mixed valence”。混价的作用导混价的作用导致在致在Fermi面面附近存在非常窄的能带附近存在非常窄的能带(部分填充部分填充f能带或能带或f能级）能级），电子可以在，电子可以在4f能级和离域化能带之间转移，对固体基态性能级和离域化能带之间转移，对固体基态性质产生显著影响。质产生显著影响。2. 窄能带现象的理论模型窄能带现象的理论模型选择经验参数的模型选择经验参数的模型Hamilton量方法量方法Hubbard模型和模型和Anderson模型模型The Hubbard ModelFrom simple quantum mechanics to many-particle interaction in solids-a short introductionHistorical factsHubbard 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 A.E.R.E., Harwell, U.K., and at the IBM Research Labs, San Jos, USA.Picture taken from: Physics Today, Vol. 34, No4, 1981What, in general, is the HM? Hubbard model is a quantum theoretical model for many-particle interaction in and with a periodic latticeIt is based on an interaction Hamitonian, some transformations and assumptions to be able to treat certain problems (e.g. magnetic behaviour and phase transitions) with solid state theoryQuantum mechanicsBasics:Schrdinger equationExpectation values Orthonormality and closure relationThe bra-ket notationBasis transformation, mathematicallyA basis transformation can be simply performed:An equation is transformed the same way:Single particle equationsParticle in a potential:Periodic potentials:Solution for weak coupling to potential: Bloch waveSingle particle equationsDispersion relation for free electrons (dashed line): Dispersion relation for Bloch electrons (quasi-free)(solid line): The energies atare no longer degenerated. Two eigenenergies at those points.Graph from Gerd Czycholl, Theoretische Festkrperphysik“, Vieweg-VerlagSingle particle equationsWannier states produce an orthonormal base of localized states; atomic wavefunctions would also be localized, but they are not orthonormal.Stronger lattice potential: coupling to lattice points occurs; a modified Bloch wave is used, e.g. Wannier states resulting from the Tight-Binding-Model:Comparison between the two new wavefunctionsBloch wavefunctionWannier wavefunction (w-part)Graph from Gerd Czycholl, Theoretische Festkrperphysik“, Vieweg-VerlagGraph from Gerd Czycholl, Theoretische Festkrperphysik“, Vieweg-VerlagWavefunction for many particlesWavefunction is not simply the product of all single particle wavefunctions; 1.Particles can not be differed2.Fermions must obey Pauli principleAnsatz: SlaterdeterminanteSecond Quantization for FermionsCreation and distruction operators create or destroy states:Second QuantizationThe operators fulfill the commutator relation:This is a must, otherwise one would disturb closure relation and orthonormality of wavefunctions described by second quantizationHamiltonian for many particlesSummation over all single particles Hamiltonians + interaction Hamiltonian:interaction potential u is the repulsive Coulomb interactionOperators in second quantizationOperators in second quantizationHamiltonian in second quantizationIs transformed like the one-particle operator A(1) and the two-particle operator A(2)Coming closer to Hubbard.Evaluation of matrix elements with Wannier wave functions:Final AssumptionsNow: only direct neighbor interactions, restriction to one band.Meaning of matrix elementst: single particle hoppingU: Hubbard-U, describes onsite-Coulomb interactionV: Nearest-neighbor (density) interactionX: conditional hopping interactionThe Hubbard Models simple Hubbard modelsimple Hubbard model extended Hubbard modelextended Hubbard modeland any combination of matrix elements.Mott-Hubbard transition, insulating (Mott) phaseCase 1: Strong coupling, U/t 1: Mott insulatingstate for a half-filled system. The density of states (available states for adding or removing particle) consits of 2 “Hubbard bands” at E0 and E0+U. The system is insulating if Efermi is between the bands. This phase is antiferromagnetic, remember the Heisenberg term.Case 2: t/U1, weak coupling: Gap disappears, density of states unchanged to simple tight-binding; the Fermi energy now lies in the band middle and the system is metallic. This transition from insulating to metallic due to changes in U/t is called Mott-Hubbard transition.Mott-Hubbard transition, metallic phaseMott-Hubbard transitionSome Examples.Lets look at the following case:2D square lattice, the band we restrict to is half filledt, U 0Antiferromagnetism for half-filling U/t1, strong coupling: Spin-spin interaction expected (direct exchange interaction, RKKY interaction, super-exchange interaction): virtual hopping is introduced, treated as perturbation. Calculation and operator relations yield as only dynamical partThis is exactly the Heisenberg Hamiltonian for antiferromagnetic exchange coupling with coupling constant J.Dependence of phases on U/t and n (where n=number of electrons/lattice site)The following graph is shown without any warranty: (Perturbation theory can not be applied in the mid region of U/t)Graph from P. Fazekas, Electron correlation and magnetismLimits of the modelThe Hamiltonian is in principle applicable for every solid state problem; often, a screened potential instead of the unscreened Coulomb potential is usedUp to now, the problem is to find calculable wavefunctions; the problem is often not analytically solvable. The advantage of the Hamiltonian, not to be restricted to very special conditions, is the disadvantage during the calculationConclusionsHubbard Model is derived from many-fermion HamiltonianIs a powerful model to describe phases in terms of interactionsThanks to Gerd Czycholl for writing the book Theoretische Festkrperphysik“Graphs taken from:Theoretische Festkrperphysik / Gerd Czycholl. - Braunschweig ; Wiesbaden : Vieweg, 2000Lecture notes on electron correlation and magnetism / Patrik Fazekas. - Singapore : World Scientific, 1999Hubbard处理干净系统的，处理干净系统的，Anderson模型则被用来处理包含模型则被用来处理包含杂质的系统。近藤杂质的系统。近藤Hamilton量：量：由于穿过离心力势垒的隧道效应所引起的由于穿过离心力势垒的隧道效应所引起的d电子共振电子共振磁性区磁性区