Nonequilibrium Many-Body Theory of Quantum Systems

A Modern Introduction

Gianluca Stefanucci, Robert van Leeuwen

• Publisher: Cambridge University Press
• Year: 2013
• Print ISBN: 9780521766173
• eBook ISBN: 9781107352070
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

The Green's function method is one of the most powerful and versatile formalisms in physics, and its nonequilibrium version has proved invaluable in many research fields. This book provides a unique, self-contained introduction to nonequilibrium many-body theory. Starting with basic quantum mechanics, the authors introduce the equilibrium and nonequilibrium Green's function formalisms within a unified framework called the contour formalism. The physical content of the contour Green's functions and the diagrammatic expansions are explained with a focus on the time-dependent aspect. Every result is derived step-by-step, critically discussed and then applied to different physical systems, ranging from molecules and nanostructures to metals and insulators. With an abundance of illustrative examples, this accessible book is ideal for graduate students and researchers who are interested in excited state properties of matter and nonequilibrium physics.

Table of contents

• Contents
• 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 otherexact 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 Momentum conservation 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
• 11.1 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 Φ functional
• 11.8 Ψ functional
• 11.9 Screened functionals
• 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 G0W0 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–Büttiker 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 Keldysh Green’s functions in the double-time plane
• 16.6.2 Time-dependent current and spectral function
• 16.6.3 Screened interaction and physical interpretation
• 16.7 Inbedding technique: how to explore the reservoirs
• 16.8 Response functions from time-propagation
• Appendices
• A Fromthe N roots of 1 to the Dirac δ-function
• B Graphical approach to permanents and determinants
• C Density matrices and probability interpretation
• 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 Fromδ-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