دانلود کتاب PHY-892 Quantum Material’s Theory, from perturbation theory to dynamical-mean field theory (lecture notes).

فهرست مطالب :

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