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1、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1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21. 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
2、电子方程单电子方程3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解，自洽求解，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
3、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：4HartreeHartree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：
4、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电子气子气53. 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：NN电子电子 体系的体系的总总波函数波函数： 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（
5、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6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
6、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。7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：8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9举例：利用举例：利用HF方程和方程和Koopmann定理研究原子的电离能定理研究原子的电离能10Nonlocal excha
7、nge termHF方法的缺点：HF方程仅考虑了交换作用，没有考虑关联，许多体系不能正确描述（金属）对原子/分子体系计算量不算大，但对固体计算量就很大：用平均的局域（交换）势替代用平均的局域（交换）势替代HF方程中的非局域交换势：方程中的非局域交换势：Xa6. Slaters Xa 方法方法11energy as a function of the one electron density, nuclear-electron attraction, electron-electron repulsionThomas-Fermi approximation for the kinetic ene
8、rgySlater approximation for the exchange energy 7. 密度泛函理论密度泛函理论(Density functional theory) (1) Thomas-Fermi-Dirac Model12Thomas-Fermi ModelThomas-Fermi model (semiclassical): 1927Electron density of a uniform electron gas: Fermi wavevector and electron kinetic energy of a uniform electron gas 13We m
9、ay assume that the kinetic energy of the electron gas depends on the local electron density:The total kinetic energy of electrons in the system is therefore a functional of electron density:Thomas-Fermi Model14(2) The Hohenberg-Kohn Theorem properties are uniquely determined by the ground-state elec
10、tron 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
11、molecules with a nondegenerate ground state, the ground-state calculate 15Proof: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
12、 allowed energies of the molecule are and the number of electrons n are specified, determined as the solutions of the electronic Schrdinger equation. 16Now we need to prove that the ground-state electron probability density the number of electrons. the external potential (except for an arbitrary add
13、itive 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.determines17the exact
14、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.18Proof:Assume thusthuswhich contradicts the giveninformation.function, the exact ground-state wave function state energy for a given H
15、amiltonian If the ground state is nondegenerate, then there is only one normalizedthat gives the exact ground19According to the variation theorem, suppose that If thenis any normalizedwell-behaved trial variation function. Now use as a trial function with the HamiltonianthenSubstituting gives20Letbe
16、 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 get21By hypothesis, the two different wave functions give the same electron. Putting and adding the above two inequalitiesdensity: yieldp
17、otentials 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 grou
18、nd-state electron probability density determines the external22probability 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 functio
19、n and energy, whichHowever, the functionalsare unknown.is also written asThe functionalindependent of the externalonispotential.23(3) The Hohenberg-kohn variational theorem“For every trial density functionthat satisfiesandfor all, the following inequality holds:, is the true groundstate energy.”Proo
20、f:Letsatisfy thatandHohenberg-Kohn theorem, determines the external potential and this in turn determines the wave functiondensity . By the,that corresponds to the .where24with Hamiltonian. According to the variation theoremLet us use the wave functionas a trial variation function for the moleculeSi
21、nce 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 ground25(4) The Kohn-Sham method If we know the ground-state electron density molecular p
22、roperties 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 ca
23、pable, 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.沈吕九沈吕九26electrons that each experience the same external potential the ground-state electro
24、n 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 s
25、ystem iswhereis the one-electron Kohn-Sham Hamiltonian. 27Thus, 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 nonin
26、teracting 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 atomic28Remember thatWith the above definitions, can be written asDefine the exchange-correlati
27、on 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
28、 evaluate and they make the main contributions29Thusbecomes.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
29、, it30ground-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 orb
30、itals are those thatwith the exchange-correlation potential defined by(If is known, its functional derivative is also known.)31The 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,
31、which handles(5) Local density approximationCompare to uniform electron gas result:32Lets compare the PZ parameterization with the uniform electron gas result for rs Hubbard U轨道序：Dudarev et al.：惩罚泛函59动力学平均场理论量子多体问题局域动力学（把点阵模型映射到自洽的量子杂质模型）冻结空间涨落，考虑局域量子涨落Hubbard模型哈密顿量单格点动力学60自洽方程Anderson杂质模型DFT-DMFT61
32、流密度泛函理论处理任意强度磁场下相互作用电子体系（1987）一套规范不变且满足连续性方程的自洽方程组交换相关能量不仅依赖于电荷密度还依赖于顺磁流密度原子分子对磁场的响应，自发磁化，磁场中的二维量子点，造新的交换相关近似62相对论性密度泛函理论量子电动力学的单粒子方程：Dirac方程Dirac-Coulomb（DC）哈密顿量Dirac-Coulomb-Breit（DCB）哈密顿量63相对论性密度泛函理论相对论情形的相对论情形的HK定理，四分量定理，四分量Dirac-Kohn-Sham（DKS）方程，数值旋量基）方程，数值旋量基组，缩并组，缩并Gaussian型旋量基组型旋量基组两分量准相对论方法
33、两分量准相对论方法Breit-Pauli近似近似ZORA近似近似有效核势（有效核势（ECP）方法）方法64密度泛函微扰理论晶格振动理论线性响应-Hessian矩阵，2n+1定理冻声方法，分子动力学谱分析方法65几何Berry位相电介质极化，介电常数偶极矩-宏观极化；流-极化变化电荷密度（波函数的模）；流（波函数的位相）零电场情况下，任意两个晶体态之间的极化变化对应着 一个几何量子位相晶格振动、铁电、压电效应、自发极化、静态介电张量、电子介电常数。不如传统的微扰理论方法普适，但实现简单、计算量小66Part II:数值方法67数值离散方法基组展开LCAO基组（Gaussian基组、数值基组）实空
34、间网格68平面波基组：从OPW到PP平面波展开正交化平面波（OPW）赝势（PP）方法经验赝势 模守恒赝势 超软赝势69Muffin-tin势场与分波方法Muffin-tin势场近似缀加平面波（APW） 格林函数方法（KKR）线性化方法LAPW LMTO分波方法的发展FP-LAPW third-generation MTO, NMTO, EMTO70平面波基组：从USPP到PAW投影缀加波（PAW）方法 赝波函数空间USPP or PAW? (VASP, ABINIT, .)71实空间网格简单直观 允许通过增加网格密度系统地控制计算收敛精度线性标度 可以方便的通过实空间域分解实现并行计算处理某些
35、特殊体系（带电体系、隧穿结。）72有限差分从微分到差分提高FD方法的计算效率对网格进行优化，如曲线网格（适应网格）和局部网格优化（复合网格）结合赝势方法 多尺度（multiscale）或预处理（preconditioning）73有限元变分方法 处理复杂的边界条件矩阵稀疏程度及带状结构往往不如有限差分好 广义的本征值问题74多分辨网格上的小波基组多分辨分析半取样（semicardinal）基组75线性标度与量子力学中的局域性“近视原理” 局域化的Wannier函数或密度矩阵绝缘体：指数衰减，能隙越大衰减越快金属：零温下按幂率衰减，在有限温度下可出现指数衰减局域区域 线性标度系数，crossov
36、er76线性标度算法分治方法 费米算符展开和费米算符投影方法直接最小化方法密度矩阵最小化轨道最小化优基组密度矩阵最小化77线性标度算法基于格林函数的递归方法 脱离轨道的（orbital-free,OF）算法 对角化以外的线性标度构造有效哈密顿量的算法 几何优化与分子动力学TDDFT78Part III:应用79物理学：强相关体系模型哈密顿量 LDA+ 电子结构：CrO2点阵动力学: 钚80化学：弱作用体系松散堆积的软物质、惰性气体、生物分子和聚合物，物理吸附、Cl+HD反应用传统的密度泛函理论处理弱作用体系一个既能产生vdW相互作用系数又能产生总关联能的非局域泛函：无缝的（seamless）方
37、法GW近似 密度泛函加衰减色散（DFdD）81生命科学：生物体系困难（尺寸问题、时间尺度） QM/MM方法（饱和原子法、冻结轨道法） 简单势能面方法线性同步过渡（LST ） 二次同步过渡（QST ）完全的分子动力学并行复制动力学（parallel replica dynamics） 超动力学（hyperdynamics, metadynamics） 温度加速的动力学（temperature accelerated dynamics ）快速蒙特卡罗（on-the-fly kineric Monte Carlo）方法82纳米和材料科学：输运性质及其他输运：非平衡态第一性原理模拟 材料力学：运动学M
38、onte Carlo（KMC）- 点阵气体和元胞自动机 - 连续方程的有限差分有限元求解83光谱学：激发态和外场系综密度泛函理论 考虑系统对称性，用求和方法计算多重态激发能多体微扰理论，GW近似Bethe-Salpeter方程 TDDFT，线性响应84一些计算软件Gaussian, DMol3, Q-Chem, ADF, SIESTA VASP, CASTEP,ABINIT, PWSCF, CPMD Octopus BigDFT855.2 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
39、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。862. 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
40、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)87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)电子散射电子散射导致某一态上有限寿命导致某一态上有限寿命(驰豫时间驰豫时间
41、)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88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89Fermi Liquid Theoryw
42、here is the number operator in momentum spaceFirst of all, What is Fermi Liquid ? If a system is described by the following Hamiltonian, it is called Fermi Liquid :(1) The theory is non-trivial because of the quartic interaction(2) However, the single-particle picture might work under someassumption
43、s90Consider a system with 4 fermion interaction. The most general Hamiltonian is :In weak coupling, some interactions are more important than the other by space argument. Take 2D electron gas as an example. In the weak coupling and low temperature limit, only electrons very close to the Fermi surfac
44、e is important.91Due to momentum conservation, not all angles are freeTake as independent parameter first. For given , the other two momenta are completely fixed. However, if , the allowed phase space is enlarged to the whole Fermi surface. Thus, we expect this kind of interaction would dominate ove
45、r the others.BCS scatteringNote that this argument is only appropriate in weak coupling limit !92On the other hand, if we take as independent parameter, there is another kind of interaction would dominate the others (by similar argument)forward scatteringThus, in weak coupling, we expect these two i
46、nteractions are the most important ones.(1) For repulsive interaction, it can be shown that the BCS interaction can be safely ignored Fermi Liquid !(2) For attractive interaction, the instability is triggered The system is better described by BCS theory !93Now we are ready to write down the Fermi Li
47、quid Theory in more familiar formIn weak coupling and at low temperature, the density is not far from the free Fermi distribution. Thus, define the density variationRewrite H in terms of 94Bare energy :It is only a constant. Will ignore it later.quasi-particle spectrum :Finally , quasi-particle inte
48、ractionsCollecting all terms, the Hamiltonian is95Now use mean-field approximation , we can solve the quasti-particle distribution self-consistently.It becomes a quadratic theory and the mean-field dispersion is : Make use of the MF dispersion , we obtain the self-consistent equation :96Simple Pictu
49、re for Fermi LiquidFree theory :Now , add in the particle-particle interaction.One quasi-particle :Many quasi-particles present :97Many interesting properties :(1) Effective mass (2) Specific heat :(3) Sound velocity :(4) Spin susceptibility :Linear temperature dependenceCompare with Curie susceptib
50、ility98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99THE PHYSICS OF
51、 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 modes100REMARKABLE PROPE
52、RTIESAbsence 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 transpo
53、rtquantities. The exponent depends on the electron-electron interaction101OUTLINEWhat 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 Tun
54、neling into a Luttinger liquidLuttinger liquid with a single impurityPhysical realizations of Luttinger liquids102LITERATURE 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
55、 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/9807366103SHORTLY ABOUT FERMI LIQUIDSLandau 1957-1959Also collective excitations occur (e.g. zero sound) at finite energi
56、esLow 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 systemRenorma
57、lization of a set of parameters (e.g. effective mass)104FERMI 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 fini
58、te life-time t3D105FERMI LIQUIDS IIIcollectiveexcitations(plasmons)single-particleexcitations12340132DISPERSION OF EXCITATIONS IN 3D 0nointeractingT = 0Finite jump in momentum distributionZZ quasi-particle weight106LIFETIME OF QUASI-PARTICLESscattering out of state kscattering into state kspinscreen
59、ed Coulomb interactionenergy conservationIn 3D an integration over angular dependence takes care of d-function Fermis golden rule yields for the lifetime tT = 0107LIFETIME OF QUASI-PARTICLES IIIn 1D k, k are scalars. Integration over k yieldsWhat about the lifetime t in 1D?formally, it divergesat sm
60、all qbut we can insert asmall cut-offAt small Ti.e., this ratio cannot bemade arbitrarily smallas in 3D108BREAKDOWN OF LANDAU THEORY IN 1D12340132DISPERSION OF EXCITATIONS IN 1D collective excitations are plasmons with (RPA)single particlegaplessplasmon COLLECTIVE AND SINGLE-PARTICLE EXCITATION NON
61、DISTINCT no longer diverges at (no angular integration over direction of as in 3D ) 109THE 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 b
62、ecomes exact when linearized branches extend from Assumptions:Only small momenta exchanges are included110TOMONAGA-LUTTINGER HAMILTONIANFree part free partinteraction fermionic annihilation/creation operatorsIntroduce right moving k 0, and left moving k 0 electrons 111TL HAMILTONIAN IIInteractions f
63、ree partinteractionbackscatteringforwardumklappforward112BOSONIZATIONBOSONIZATION: 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
64、CHARGEEXCITATIONS, OR BOSONIC ELECTRON-HOLE EXCITATIONS113STEP 1WHICH OPERATORS DO THE JOB?Introduce the density operators (create excitation of momentum q)and consider their commutation relations nearly bosonic commutation relations 114STEP 1: PROOFConsider e.g.algebra offermionic operatorsoccupati
65、on operator115STEP 2Examine nowBOSONIZED HAMILTONIANSTATES CREATED BY ARE EIGENSTATES OF WITH ENERGY andinteractions116STEP 2: PROOFExample:117STEP 3Introduce the bosonic operatorsyieldingDIAGONALIZATION118SPIN-CHARGE SEPARATIONand interaction (satisfying SU2 symmetry)If we include spin, it gets sli
66、ghtly more complicated . and interesting Introduce the spin and charge densitiesHamiltonian decouple in two independent spin and charge parts,with excitations propagating with velocities 119SPACE REPRESENTATIONLong wavelength limit (interactions )Appropriate linear combinations P, q of the field (x)
67、 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。1284f，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YSc129Smith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130另一类窄带现象：来自能带中的近
68、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，而三者之一来描述，而应该说三者同时发挥作用。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”。混价的作用导致。混价的作用导致在在Fermi面附近存在非常窄的能带面附近存在非常窄的能带(部
69、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，对固体基态性质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。产生显著影响。1312. 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132The Hubbard ModelFrom simple quantum mechanics to many-particle interaction in solids-a short introduction133Historical facts
70、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 A.E.R.E., Harwell, U.K., and at the IBM Research Labs, San
71、Jos, USA.Picture taken from: Physics Today, Vol. 34, No4, 1981134What, 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 ce
72、rtain problems (e.g. magnetic behaviour and phase transitions) with solid state theory135Quantum mechanicsBasics:Schrdinger equationExpectation values Orthonormality and closure relationThe bra-ket notation136Basis transformation, mathematicallyA basis transformation can be simply performed:An equat
73、ion is transformed the same way:137Single particle equationsParticle in a potential:Periodic potentials:Solution for weak coupling to potential: Bloch wave138Single particle equationsDispersion relation for free electrons (dashed line): Dispersion relation for Bloch electrons (quasi-free)(solid line
74、): The energies atare no longer degenerated. Two eigenenergies at those points.Graph from Gerd Czycholl, Theoretische Festkrperphysik“, Vieweg-Verlag139Single particle equationsWannier states produce an orthonormal base of localized states; atomic wavefunctions would also be localized, but they are
75、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:140Comparison between the two new wavefunctionsBloch wavefunctionWannier wavefunction (w-part)Graph from Gerd Czycholl, Theoretisch
76、e Festkrperphysik“, Vieweg-VerlagGraph from Gerd Czycholl, Theoretische Festkrperphysik“, Vieweg-Verlag141Wavefunction for many particlesWavefunction is not simply the product of all single particle wavefunctions; 1.Particles can not be differed2.Fermions must obey Pauli principleAnsatz: Slaterdeter
77、minante142Second Quantization for FermionsCreation and distruction operators create or destroy states:143Second QuantizationThe operators fulfill the commutator relation:This is a must, otherwise one would disturb closure relation and orthonormality of wavefunctions described by second quantization1
78、44Hamiltonian for many particlesSummation over all single particles Hamiltonians + interaction Hamiltonian:interaction potential u is the repulsive Coulomb interaction145Operators in second quantization146Operators in second quantization147Hamiltonian in second quantizationIs transformed like the on
79、e-particle operator A(1) and the two-particle operator A(2)148Coming closer to Hubbard.Evaluation of matrix elements with Wannier wave functions:150Final AssumptionsNow: only direct neighbor interactions, restriction to one band.151Meaning of matrix elementst: single particle hoppingU: Hubbard-U, de
80、scribes onsite-Coulomb interactionV: Nearest-neighbor (density) interactionX: conditional hopping interaction152The Hubbard Models simple Hubbard modelsimple Hubbard model extended Hubbard modelextended Hubbard modeland any combination of matrix elements.153Mott-Hubbard transition, insulating (Mott)
81、 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, remembe
82、r the Heisenberg term.154Case 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 transitionMott-
83、Hubbard transition.Mott-Hubbard transition, metallic phase155Mott-Hubbard transition156Some Examples.Lets look at the following case:2D square lattice, the band we restrict to is half filledt, U 0157Antiferromagnetism for half-filling U/t1, strong coupling: Spin-spin interaction expected (direct exc
84、hange 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.158Depe
85、ndence 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 magnetism159Limits of the modelThe Hamiltonian is in principle a
86、pplicable 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 c
87、onditions, is the disadvantage during the calculation160ConclusionsHubbard 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 Festkrperph
88、ysik / Gerd Czycholl. - Braunschweig ; Wiesbaden : Vieweg, 2000Lecture notes on electron correlation and magnetism / Patrik Fazekas. - Singapore : World Scientific, 1999161Hubbard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：162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163磁性区磁性区164
