Contents: A refresher in statistical mechanics and quantum mechanics; Statistical Physics and Density matrix;
Second quantization;Hartree-Fock approximation; Model Hamiltonians; Broken symmetry and canonical transformations; Elementary quantum mechanics and path integrals; Correlation functions, general properties; Relation between correlation functions and experiments; Time-dependent perturbation theory; Linear-response theory; General properties of correlation functions; Kubo formula for the conductivity; Drude weight, metals, insulators and superconductors; Relation between conductivity and dielectric constant; Introduction to Green’s functions. One-body Schrödinger equation; Definition of the propagator, or Green’s function; Information contained in the one-body propagator; A first phenomenological encounter with self-energy; Perturbation theory for one-body propagator; Formal properties of the self-energy; Electrons in a random potential: Impurity averaging technique; Other perturbation resummation techniques: a preview; Feynman path integral for the propagator, and alternate formu-
lation of quantum mechanics; The one-particle Green’s function at finite temperature; Motivation for the definition of the second quantized Green’s function; Interaction representation, when time order matters; Kadanoff-Baym and Keldysh-Schwinger contours; Matsubara Green’s function and its relation to usual Green’s functions; Physical meaning of the spectral weight: Quasiparticles, effective mass, wave function renormalization, momentum distribution; A few more formal matters : asymptotic behavior and causality; Three general theorems; The Coulomb gas; Feynman rules for two-body interactions; First step with functional derivatives: Hartree-Fock and RPA; Collective modes in non-interacting limit; Interactions and collective modes in a simple way; Density response in the presence of interactions; More formal matters: Consistency relations between single-particle self-energy, collective modes, potential energy and free energy; Single-particle properties and Hartree-Fock; Second step of the approximation: GW curing Hartree-Fock theory; Physics in single-particle properties; Beyond RPA: skeleton diagrams, vertex functions and associated difficulties; Fermions on a lattice: Hubbard and Mott; Density functional theory; The Hubbard model in the footsteps of the electron gas; Dynamical Mean-Field Theory and Mott transition; Broken Symmetry; Antiferromagnetism close to half-filling and pseudogap in two dimensions; Feynman’s derivation of the thermodynamic variational principle
for quantum systems; Hubbard-Stratonovich transformation and
critical phenomena. |