نظریه مواد کوانتومی PHY-892: از نظریه اختلال تا نظریه میدان میانگین دینامیکی (یادداشت‌های درسی)

A reminder of some de...nitions and theorems on Fourier transforms and integrals of functions of a complex number . . . . . 2.3 The e¤ect of damping can be retarded. Where we encounter the consequences of causality and the Kramers-Kronig relations . . . . 3 Susceptibility, propagator: some general properties 3.1 De...nition of the susceptibility and preview of some of its properties 3.2 Real-time version of the retarded susceptibility . . . . . . . . . . . 3.3 Positivity of the power absorbed, implies that ! 00 (!) is positive . 4 Dissipation and irreversibility emerge in the limit of an in...nite number of degrees of freedom 4. Preamble Aim In broad strokes, adiabatic continuity and broken symmetry Contents Summary I Classical hamonic oscillator to introduce basic mathematical tools and concepts The damped, driven, harmonic oscillator The driven harmonic oscilator Interlude: A reminder of some definitions and theorems on Fourier transforms and integrals of functions of a complex number The effect of damping can be retarded. Where we encounter the consequences of causality and the Kramers-Kronig relations Susceptibility, propagator: some general properties Definition of the susceptibility and preview of some of its properties Real-time version of the retarded susceptibility Positivity of the power absorbed, implies that 0=x"01210=x"011F( 0=x"0121) is positive Dissipation and irreversibility emerge in the limit of an infinite number of degrees of freedom Example of an oscillator attached to a bath of harmonic oscillators: A model in the Caldeira-Leggett category Irreversibity emerges in the limit of an infinite bath *Fluctuations and dissipation are related *Fluctuations may also be seen as generated from fluctuating internal forces. and sum-rules come out naturally. The Langevin approach Irreducible self-energy and virtual particules in a (almost) classical context The concept of self-energy emerges naturally when one does a power series expansion Virtual particles Exercices for Part I Devoir 2, fonctions de réponse, théorème de Kramers Kronig II Correlation functions, general properties Relation between correlation functions and experiments Quite generally, Fermi's golden rule in either scattering or relaxation experiments lead observables that are time-dependent correlation functions *Details of the derivation for the specific case of electron scattering Time-dependent perturbation theory Schrödinger and Heisenberg pictures. Interaction picture and perturbation theory Linear-response theory General properties of correlation functions Notations and definition of 0=x"011F Symmetry properties of H and symmetry of the response functions Translational invariance *Parity Time-reversal symmetry in the absence of spin is represented by complex conjugation for the wave function and by the transpose for operators *Time-reversal symmetry in the presence of spin necessitates a matrix representation Properties that follow from the definition and proof that 0=x"011F0=x"011Aq0=x"011A-q(0=x"0121)=-0=x"011F0=x"011Aq0=x"011A-q(-0=x"0121) Kramers-Kronig relations follow from causality Spectral representation and Kramers-Kronig relations *Positivity of 0=x"01210=x"011F(0=x"0121) and dissipation A short summary of basic symmetry properties and constraints on 0=x"011F *Fluctuation-dissipation theorem Lehmann representation and spectral representation Sum rules Thermodynamic sum-rules The order of limits when 0=x"0121 or q tends to zero is important for 0=x"011F Moments, sum rules, and their relation to high-frequency expansions. The f-sum rule as an example Kubo formulae for the conductivity Coupling between electromagnetic fields and matter, and gauge invariance *Invariant action, Lagrangian and coupling of matter and electromagnetic field *Lagrangian for the electromagnetic field Response of the current to external vector and scalar potentials Kubo formula for the transverse conductivity Kubo formula for the longitudinal conductivity and f-sum rule A gauge invariant expression for the longitudinal conductivity that follows from current conservation Further consequences of gauge invariance and relation to f sum-rule. Longitudinal conductivity sum-rule and a useful expression for the longitudinal conductivity. Drude weight, metals, insulators and superconductors The Drude weight What is a metal What is an insulator What is a superconductor Metal, insulator and superconductor, a summary Finding the London penetration depth from optical conductivity *Relation between conductivity and dielectric constant *Transverse dielectric constant. Longitudinal dielectric constant Exercices for part II Lien entre fonctions de réponses, constante de diffusion et dérivées thermodynamiques. Rôle des règles de somme. Fonction de relaxation de Kubo. Constante diélectrique et Kramers-Kronig. III Introduction to Green's functions. One-body Schrödinger equation Definition of the propagator, or Green's function Preliminaries: some notation Definition of the Green's function and physical meaning *The initial condition can be at some arbitrary time Various ways of representing the one-body propagator, their properties and the information they contain Representation in frequency space and Lehmann representation *Operator representation in frequency space Observables can be obtained from the Green's function *Spectral representation, Kramers-Kronig, sum rules and high frequency expansion Spectral representation and Kramers-Kronig relations. *Sum rules *High frequency expansion. *Relation to transport and fluctuations A first phenomenological encounter with self-energy *Perturbation theory for one-body propagator Perturbation theory in operator form Feynman diagrams for a one-body potential and their physical interpretation. A basis with plane wave states normalized to unity Diagrams in position space Diagrams in momentum space Dyson's equation, irreducible self-energy *Formal properties of the self-energy *Electrons in a random potential: Impurity averaging technique. *Impurity averaging *Averaging of the perturbation expansion for the propagator *Other perturbation resummation techniques: a preview *Feynman path integral for the propagator, and alternate formulation of quantum mechanics *Physical interpretation *Computing the propagator with the path integral Exercices for part III Fonctions de Green retardées, avancées et causales. Partie imaginaire de la self-énergie et règle d'or de Fermi Règles de somme dans les systèmes désordonnés. Développement du locateur dans les systèmes désordonnés. Une impureté dans un réseau: état lié, résonnance, matrice T. Diffusion sur des impuretés. Résistance résiduelle des métaux. IV The one-particle Green's function at finite temperature Main results from second quantization Fock space, creation and annihilation operators Creation-annihilation operators for fermion wave functions Creation-annihilation operators for boson wave functions Number operator and normalization Change of basis General case The position and momentum space basis Wave functions One-body operators Number operator and the nature of states in second quantization Going backwards from second to first quantization Two-body operators. Getting familiar with second quantized operators in the Heisenberg picture, commutator identities *Formal derivation of second quantization *A quantization recipe *Applying the quantization recipe to wave equations Motivation for the definition of the second quantized Green's function GR Measuring a two-point correlation function (ARPES) Definition of the many-body GR and link with the previous one Examples with quadratic Hamiltonians: Spectral representation of GR and analogy with susceptibility Interaction representation, when time order matters *Kadanoff-Baym and Keldysh-Schwinger contours Matsubara Green's function and its relation to usual Green's functions. (The case of fermions) Definition for fermions Time ordered product in practice Antiperiodicity and Fourier expansion (Matsubara frequencies) * GR and G can be related using contour integration The Lehmann representation tells us the physical meaning of the spectral weight and the relation between GR and G Spectral weight and rules for analytical continuation Matsubara Green's function for translationally invariant systems Matsubara Green's function in the non-interacting case G0( k;ikn) from the spectral representation *G0( k;0=x"011C) and G0( k;ikn) from the definition *G0( k;0=x"011C) and G0( k;ikn) from the equations of motion Sums over Matsubara frequencies Susceptibility and linear response in Matsubara space Matsubara frequencies for the susceptibility, as bosonic correlation function Linear response in imaginary time Physical meaning of the spectral weight: Quasiparticles, effective mass, wave function renormalization, momentum distribution. Probabilistic interpretation of the spectral weight Analog of the fluctuation dissipation theorem Some experimental results from ARPES Quasiparticles Fermi liquid interpretation of ARPES Momentum distribution in an interacting system *More formal matters : asymptotic behavior, causality, gauge transformation *Asymptotic behavior of G( k;ikn) and ( k;ikn) *Implications of causality for GR and R Gauge transformation for the Green's function Three general theorems Wick's theorem Linked cluster theorems Linked cluster theorem for normalized averages Linked cluster theorem for characteristic functions or free energy Variational principle and application to Hartree-Fock theory Thermodynamic variational principle Thermodynamic variational principle for classical systems based on the linked-cluster theorem Application of the variational principle to Hartree-Fock theory Exercices for part IV Bosonic Matsubara frequencies. First quantization from the second Retrouver la première quantification à partir de la seconde Non interacting Green's function from the spectral weight and analytical continuation Sum over bosonic Matsubara frequencies Représentation de Lehman et prolongement analytique Représentation de Lehman et prolongement analytique pour les fermions Fonction de Green pour les phonons Oscillateur harmonique en contact avec un réservoir Limite du continuum pour le réservoir, et irréversibilité V The Coulomb gas The functional derivative approach External fields to compute correlation functions Green's functions and higher order correlations from functional derivatives Source fields for Green's functions, an impressionist view Equations of motion to find G in the presence of source fields Hamiltonian and equations of motion for 0=x"0120( 1) Equations of motion for G0=x"011E and definition of 0=x"011E Four-point function from functional derivatives Self-energy from functional derivatives The self-energy, one-particle irreducibility and Green's function First steps with functional derivatives: Hartree-Fock and RPA Functional derivatives can be used to generate perturbation theory Skeleton expansion Expansion in terms of the bare Green's function Hartree-fock and RPA in space-time Hartree-Fock and RPA in Matsubara and momentum space with 0=x"011E=0 *Feynman rules for two-body interactions Hamiltonian and notation *In position space *Proof of the overall sign of a Feynman diagram In momentum space *Feynman rules for the irreducible self-energy *Feynman diagrams and the Pauli exclusion principle Particle-hole excitations in the non-interacting limit and the Lindhard function Definitions and analytic continuation Density response in the non-interacting limit in terms of G0=x"011B0 *The Feynman way The Schwinger way (source fields) Density response in the non-interacting limit: Lindhard function Zero-temperature value of the Lindhard function: the particle-hole continuum Interactions and collective modes in a simple way Expansion parameter in the presence of interactions: rs Thomas-Fermi screening Reducible and irreducible susceptibilities: another look at the longitudinal dielectric constant Plasma oscillations Density response in the presence of interactions Density-density correlations, RPA *The Feynman way The Schwinger way Explicit form for the dielectric constant and special cases Particle-hole continuum Screening Friedel oscillations Plasmons and Landau Damping f-sum rule Single-particle properties and Hartree-Fock *Variational approach Hartree-Fock from the point of view of Green's functions Hartree-Fock from the point of view of renormalized perturbation theory and effective medium theories The pathologies of the Hartree-Fock approximation for the electron gas. *More formal matters: Consistency relations between single-particle self-energy, collective modes, potential energy and free energy *Consistency between self-energy and density fluctuations *Equations of motion for the Feynmay way Self-energy, potential energy and density fluctuations Second step of the approximation: GW curing Hartree-Fock theory *An approximation for that is consistent with the Physics of screening Self-energy and screening, GW the Schwinger way Physics in single-particle properties Single-particle spectral weight Simplifying the expression for Physical processes contained in Fermi liquid results Comparison with experiments Free-energy calculations Free energy and consistency between one and two-particle quantities Free energy for the Coulomb gas in the RPA approximation Landau Fermi liquid for response functions Compressibility *Expansion in terms of dressed or bare Green's functions: Skeleton diagrams The expansion in terms of bare Green's functions can be derived using the Schwinger approach *General considerations on perturbation theory and asymptotic expansions *Beyond RPA: skeleton diagrams, vertex functions and associated difficulties. *A dressed bubble diagram violates charge conservation *RPA with dressed bubble violates the f-sum rule and gives bad results *Two reformulations of perturbation theory *Skeleton diagrams *Channels *Crossing symmetry *Hedin's equations Exercices for part V Théorie des perturbations au deuxième ordre pour la self-énergie Théorie des perturbations au deuxième ordre pour la self-énergie à la Schwinger Déterminant, théorème de Wick et fonctions à plusieurs points dans le cas sans interaction Determinant, Wick's theorem and many-point correlation functions in the non-interacting case Cas particulier du théorème de Wick avec la méthode de Schwinger Fonction de Lindhard et susceptibilité magnétique: VI Fermions on a lattice: Hubbard and Mott Density functional theory The ground state energy is a functional of the local density The Kohn-Sham approach *Finite temperature Improving DFT with better functionals DFT and many-body perturbation theory Model Hamiltonians The Hubbard model Assumptions behind the Hubbard model Where spin fluctuations become important The non-interacting limit U=0 The strongly interacting, atomic, limit t=0 *The Peierls substitution allows one to couple general tight-binding models to the electromagnetic field The Hubbard model in the footsteps of the electron gas Single-particle properties Response functions Hartree-Fock and RPA RPA and violation of the Pauli exclusion principle Why RPA violates the Pauli exclusion principle from the point of view of diagrams RPA, phase transitions and the Mermin-Wagner theorem The Two-Particle-Self-Consistent approach TPSC First step: two-particle self-consistency for G( 1) ,( 1) , sp( 1) =Usp and ch( 1) =Uch TPSC Second step: an improved self-energy ( 2) TPSC, internal accuracy checks TPSC, benchmarking and physical aspects Physically motivated approach, spin and charge fluctuations Mermin-Wagner, Kanamori-Brueckner Benchmarking Spin and charge fluctuations Self-energy TPSC+, Beyond TPSC *Antiferromagnetism close to half-filling and pseudogap in two dimensions Pseudogap in the renormalized classical regime Pseudogap in electron-doped cuprates Dynamical Mean-Field Theory and Mott transition-I A simple example of a model exactly soluble by mean-field theory Mean-field theory in classical physics The self-energy is independent of momentum in infinite dimension The dynamical mean-field self-consistency relation, derivation 1 Quantum impurities: The Anderson impurity model The dynamical mean-field self-consistency relation, derivation 2 Perturbation theory for the Anderson impurity model is the same as before but with a Green's function that contains the hybridization function. The Mott transition Doped Mott insulators A short history Model Hamiltonians Early work Solving the Hubbard Hamiltonian in infinite dimension Dynamical Mean-Field Theory (DMFT) Single-Site Dynamical Mean-Field Theory Cluster generalizations of DMFT Impurity solvers Merging DFT and Dynamical Mean-Field Theory Exercices for part VI Symétrie particule-trou pour Hubbard Règle de somme f Impureté quantique dans le cas sans interaction Screening of spin fluctuations by the Coulomb interaction: Generalized RPA: Atomic limit ( t=0) :. Limite atomique ( t=0) : VII Broken Symmetry Some general ideas on the origin of broken symmetry Instability of the normal state The noninteracting limit and rotational invariance Effect of interactions for ferromagnetism, the Schwinger way *Effect of interactions for ferromagnetism, the Feynman way The thermodynamic Stoner instability Magnetic structure factor and paramagnons Weak interactions at low filling, Stoner ferromagnetism and the Broken Symmetry phase Simple arguments, the Stoner model Variational wave function Feynman's variational principle for variational Hamiltonian. Order parameter and ordered state The mean-field Hamiltonian can be obtained by a method where the neglect of fluctuations is explicit The gap equation and Landau theory in the ordered state The Green function point of view (effective medium) There are residual interactions Collective Goldstone mode, stability and the Mermin-Wagner theorem Tranverse susceptibility Thermodynamics and the Mermin-Wagner theorem Kanamori-Brückner screening: Why Stoner ferromagnetism has problems Exercices Antiferromagnétisme itinérant *Additional remarks: Hubbard-Stratonovich transformation and critical phenomena Electron-phonon interactions in metals (jellium) Beyond the Born-Oppenheimer approximation, electron-phonon interaction, Kohn anomaly Hamiltonian and matrix elements for interactions in the jellium model Place holder Dielectric constant for mobile ions The plasmon frequency of the ions is replaced by an acoustic mode due to screening Effective electron-electron interaction mediated by phonons RPA approximation Effective mass, quasiparticle renormalization, Kohn anomaly and Migdal's theorem Instability of the normal phase in the Schwinger formalism Nambu space and generating functional Equations of motion Pair susceptibility BCS theory Broken symmetry, analogy with the ferromagnet The BCS equation the Green's function way (effective medium) Phase coherence Eliashberg equation Hamiltonien BCS réduit Méthode de diagonalisation utilisant l'algèbre des spineurs Approche variationnelle Cohérence de phase, fonction d'onde Singlet s-wave superconductivity Solution de l'équation BCS, Tc et équation de Ginzburg-Landau, gap à T=0 s,p,d... symmetries in the solution of the BCS equation Exercices for part VII Principe variationel et ferromagnétisme de Stoner: Antiferromagnétisme itinérant Supraconductivité: conductivité infinie et effet Meissner: Principe variationnel à T=0 pour le ferroaimant Équations de champ moyen pour le ferroaimant Variational principle at T=0 for the ferromagnet Mean-field equations for the ferromagnet VIII Advanced topics: Coherent state functional integral, Luttinger Ward etc Luttinger-Ward functional The self-energy can be expressed as a functional derivative with respect to the Green's function The free energy of a non-interacting but time-dependent problem is -TTr[ ln( -G-1-G-1) ] The Luttinger-Ward functional and the Legendre transform of -TlnZ[ 0=x"011E] * Formal matters: recipes to satisfy conservation laws *Ward-Takahashi identity for charge conservation *The Ward identity from gauge invariance *Particle-number conservation is garanteed if is obtained from 0=x"010E[G]/0=x"010EG *Other formal consequences of [G] *Thermodynamic consistency * Luttinger's theorem Conserving approximations are not a panacea The constraining field method Another derivation of the Baym-Kadanoff functional The Luttinger-Ward functional can be written in terms of two-particle irreducible skeleton diagrams A non-perturbative approach based on the constraining field vs the skeleton expansion The self-energy functional approach and DMFT The self-energy functional Variational cluster perturbation theory, or variational cluster approximation Cellular dynamical mean-field theory The Dynamical cluster approximation Coherent-states for bosons Coherent states for fermions Grassmann variables for fermions Grassmann integrals Change of variables in Grassmann integrals Grassmann Gaussian integrals Closure, overcompleteness and trace formula The coherent state functional integral for fermions A simple example for a single fermion without interactions Generalization to a continuum and to a time dependent one-body Hamiltonian Wick's theorem *Source fields and Wick's theorem Interactions and quantum impurities as an example IX Many-body in a nutshell Handeling many-interacting particles: second quantization Fock space, creation and annihilation operators Change of basis The position and momentum space basis Wave function One-body operators Two-body operators. The Hubbard model to illustrate some of the concepts The Hubbard model Perturbation theory and time-ordered products Green functions contain useful information Photoemission experiments and fermion correlation functions Definition of the Matsubara Green function The Matsubara frequency representation is convenient Spectral weight and how it is related to Gk( ikn) and to photoemission Gk( ikn) for the non-interacting case U=0 Obtaining the spectral weight from Gk( ikn) : the problem of analytic continuation Self-energy and the effect of interactions The atomic limit, t=0 The self-energy and the atomic limit example (Mott insulators) A few properties of the self-energy Integrating out the bath in the quantum-impurity problem: The Anderson impurity model Many-particle correlation functions and Wick's theorem Source fields to calculate many-body Green functions A simple example in classical statistical mechanics Green functions and higher order correlations from source fields Equations of motion to find G0=x"011E and 0=x"011E Hamiltonian and equations of motion for 0=x"0120( 1) Equations of motion for G0=x"011E and definition of 0=x"011E The general many-body problem An integral equation for the four-point function Self-energy from functional derivatives Long-range forces and the GW approximation Equations in space-time Equations in momentum space with 0=x"011E=0 Density response in the RPA Self-energy and screening in the GW approximation Hedin's equations Luttinger-Ward functional and related functionals A glance at coherent state functional integrals Fermion coherent states Grassmann calculus Recognizing the Hamiltonian in the action X Appendices Statistical Physics and Density matrix Density matrix in ordinary quantum mechanics Density Matrix in Statistical Physics Legendre transforms Legendre transform from the statistical mechanics point of view Second quantization Describing symmetrized or antisymmetrized states Change of basis Second quantized version of operators One-body operators Two-body operators Hartree-Fock approximation The theory of everything Variational theorem Wick's theorem Minimization and Hartree-Fock equations Model Hamiltonians Heisenberg and t-J model Anderson lattice model Broken symmetry and canonical transformations The BCS Hamiltonian Feynman's derivation of the thermodynamic variational principle for quantum systems Definitions