Preface List of abbreviations and acronyms Fundamental constants and basic relations 1 Second quantization 1.1 Quantum mechanics of one particle 1.2 Quantum mechanics of many particles 1.3 Quantum mechanics of many identical particles 1.4 Field operators 1.5 General basis states 1.6 Hamiltonian in second quantization 1.7 Density matrices and quantum averages 2 Getting familiar with second quantization: model Hamiltonians 2.1 Model Hamiltonians 2.2 Pariser-Parr-Pople model 2.3 Noninteracting models 2.3.1 Bloch theorem and band structure 2.3.2 Fano model 2.4 Hubbard model 2.4.1 Particle-hole symmetry: application to the Hubbard dimer 2.5 Heisenberg model 2.6 BCS model and the exact Richardson solution 2.7 Holstein model 2.7.1 Peierls instability 2.7.2 Lang-Firsov transformation: the heavy polaron 3 Time-dependent problems and equations of motion 3.1 Introduction 3.2 Evolution operator 3.3 Equations of motion for operators in the Heisenberg picture 3.4 Continuity equation: paramagnetic and diamagnetic currents 3.5 Lorentz Force 4 The contour idea 4.1 Time-dependent quantum averages 4.2 Time-dependent ensemble averages 4.3 Initial equilibrium and adiabatic switching 4.4 Equations of motion on the contour 4.5 Operator correlators on the contour 5 Many-particle Green's functions 5.1 Martin-Schwinger hierarchy 5.2 Truncation of the hierarchy 5.3 Exact solution of the hierarchy from Wick's theorem 5.4 Finite and zero-temperature formalism from the exact solution 5.5 Langreth rules 6 One-particle Green's function 6.1 What can we learn from G? 6.1.1 The inevitable emergence of memory 6.1.2 Matsubara Green's function and initial preparations 6.1.3 Lesser/greater Green's function: relaxation and quasi-particles 6.2 Noninteracting Green's function 6.2.1 Matsubara component 6.2.2 Lesser and greater components
6.2.3 All other components and a useful exercise 6.3 Interacting Green's function and Lehmann representation 6.3.1 Steady-states, persistent oscillations, initial-state dependence 6.3.2 Fluctuation-dissipation theorem and other exact properties 6.3.3 Spectral function and probability interpretation 6.3.4 Photoemission experiments and interaction effects 6.4 Total energy from the Galitskii-Migdal formula 7 Mean field approximations 7.1 Introduction 7.2 Hartree approximation 7.2.1 Hartree equations 7.2.2 Electron gas 7.2.3 Quantum discharge of a capacitor 7.3 Hartree-Fock approximation 7.3.1 Hartree-Fock equations 7.3.2 Coulombic electron gas and spin-polarized solutions 8 Conserving approximations: two-particle Green's function 8.1 Introduction 8.2 Conditions on the approximate G2 8.3 Continuity equation 8.4 Momentumconservation law 8.5 Angular momentum conservation law 8.6 Energy conservation law 9 Conserving approximations: self-energy 9.1 Self-energy and Dyson equations I 9.2 Conditions on the approximate Σ 9.3 φ functional 9.4 Kadanoff-Baym equations 9.5 Fluctuation-dissipation theorem for the self-energy 9.6 Recovering equilibrium from the Kadanoff-Baym equations 9.7 Formal solution of the Kadanoff-Baym equations 10 MBPT for the Green's function 10.1 Getting started with Feynman diagrams 10.2 Loop rule 10.3 Cancellation of disconnected diagrams 10.4 Summing only the topologically inequivalent diagrams 10.5 Self-energy and Dyson equations II 10.6 G-skeleton diagrams 10.7 W-skeleton diagrams 10.8 Summary and Feynman rules 11 MBPT and variational principles for the grand potential 1l.l Linked cluster theorem 11.2 Summing only the topologically inequivalent diagrams 11.3 How to construct the φ functional 11.4 Dressed expansion of the grand potential 11.5 Luttinger-Ward and Klein functionals 11.6 Luttinger-Ward theorem 11.7 Relation between the reducible polarizability and the ~ functional
12 MBPT for the two-particle Green's function 12.1 Diagrams for G2 and loop rule 12.2 Bethe-Salpeter equation 12.3 Excitons 12.4 Diagrammatic proof of K = ±δΣ/δG 12.5 Vertex function and Hedin equations 13 Applications of MBPT to equilibrium problems 13.1 Lifetimes and quasi-particles 13.2 Fluctuation-dissipation theorem for P and W 13.3 Correlations in the second-Born approximation 13.3.1 Polarization effects 13.4 Ground-state energy and correlation energy 13.5 GW correlation energy of a Coulombic electron gas 13.6 T-matrix approximation 13.6.1 Formation of a Cooper pair 14 Linear response theory: preliminaries 14.1 Introduction 14.2 Shortcomings of the linear response theory 14.2.1 Discrete-discrete coupling 14.2.2 Discrete-continuum coupling 14.2.3 Continuum-continuum coupling 14.3 Fermi golden rule 14.4 Kubo formula 15 Linear response theory: many-body formulation 15.1 Current and density response function 15.2 Lehmann representation 15.2.1 Analytic structure 15.2.2 The f-sum rule 15.2.3 Noninteracting fermions 15.3 Bethe-Salpeter equation from the variation of a conserving G 15.4 Ward identity and the f-sum rule 15.5 Time-dependent screening in an electron gas 15.5.1 Noninteracting density response function 15.5.2 RPA density response function 15.5.3 Sudden creation of a localized hole 15.5.4 Spectral properties in the GoWo approximation 16 Applications of MBPT to nonequilibrium problems 16.1 Kadanoff-Baym equations for open systems 16.2 Time-dependent quantum transport: an exact solution 16.2.1 Landauer-B?ttiker formula 16.3 Implementation of the Kadanoff-Baym equations 16.3.1 Time-stepping technique 16.3.2 Second-Born and GW self-energies 16.4 Initial-state and history dependence 16.5 Charge conservation 16.6 Time-dependent GW approximation in open systems 16.6.1 16.8 Response functions from time-propagation Appendices A From the N roots of ! to the Dirac δ-function B Graphical approach to permanents and determinants C Density matrices and probability interpretatio D Thermodynamics and statistical mechanics E Green's functions and lattice symmetry F Asymptotic expansions G Wick's theorem for general initial states H BBGKY hierarchy I From δ-like peaks to continuous spectral functions J Virial theorem for conserving approximations K Momentum distribution and sharpness of the Fermi surface L Hedin equations from a generating functional M Lippmann-Schwinger equation and cross-section N Why the name Random Phase Approximation~ O Kramers-Kronig relations P Algorithm for solving the Kadanoff-Baym equations References Index
[