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Nonequilibrium Many-Body Theory of Quantum Systems

Book Description

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

  1. Cover
  2. Half Title
  3. Title
  4. Copyright
  5. Table of Content
  6. Preface
  7. List of abbreviations and acronyms
  8. Fundamental constants and basic relations
  9. 1 Second quantization
    1. 1.1 Quantum mechanics of one particle
    2. 1.2 Quantum mechanics of many particles
    3. 1.3 Quantum mechanics of many identical particles
    4. 1.4 Field operators
    5. 1.5 General basis states
    6. 1.6 Hamiltonian in second quantization
    7. 1.7 Density matrices and quantum averages
  10. 2 Getting familiar with second quantization: model Hamiltonians
    1. 2.1 Model Hamiltonians
    2. 2.2 Pariser–Parr–Pople model
    3. 2.3 Noninteracting models
      1. 2.3.1 Bloch theorem and band structure
      2. 2.3.2 Fano model
    4. 2.4 Hubbard model
      1. 2.4.1 Particle–hole symmetry: application to the Hubbard dimer
    5. 2.5 Heisenberg model
    6. 2.6 BCS model and the exact Richardson solution
    7. 2.7 Holstein model
      1. 2.7.1 Peierls instability
      2. 2.7.2 Lang–Firsov transformation: the heavy polaron
  11. 3 Time-dependent problems and equations of motion
    1. 3.1 Introduction
    2. 3.2 Evolution operator
    3. 3.3 Equations of motion for operators in the Heisenberg picture
    4. 3.4 Continuity equation: paramagnetic and diamagnetic currents
    5. 3.5 Lorentz Force
  12. 4 The contour idea
    1. 4.1 Time-dependent quantum averages
    2. 4.2 Time-dependent ensemble averages
    3. 4.3 Initial equilibrium and adiabatic switching
    4. 4.4 Equations of motion on the contour
    5. 4.5 Operator correlators on the contour
  13. 5 Many-particle Green’s functions
    1. 5.1 Martin–Schwinger hierarchy
    2. 5.2 Truncation of the hierarchy
    3. 5.3 Exact solution of the hierarchy from Wick’s theorem
    4. 5.4 Finite and zero-temperature formalism from the exact solution
    5. 5.5 Langreth rules
  14. 6 One-particle Green’s function
    1. 6.1 What can we learn from G?
      1. 6.1.1 The inevitable emergence of memory
      2. 6.1.2 Matsubara Green’s function and initial preparations
      3. 6.1.3 Lesser/greater Green’s function: relaxation and quasi-particles
    2. 6.2 Noninteracting Green’s function
      1. 6.2.1 Matsubara component
      2. 6.2.2 Lesser and greater components
      3. 6.2.3 All other components and a useful exercise
    3. 6.3 Interacting Green’s function and Lehmann representation
      1. 6.3.1 Steady-states, persistent oscillations, initial-state dependence
      2. 6.3.2 Fluctuation–dissipation theorem and other exact properties
      3. 6.3.3 Spectral function and probability interpretation
      4. 6.3.4 Photoemission experiments and interaction effects
    4. 6.4 Total energy from the Galitskii–Migdal formula
  15. 7 Mean field approximations
    1. 7.1 Introduction
    2. 7.2 Hartree approximation
      1. 7.2.1 Hartree equations
      2. 7.2.2 Electron gas
      3. 7.2.3 Quantum discharge of a capacitor
    3. 7.3 Hartree–Fock approximation
      1. 7.3.1 Hartree–Fock equations
      2. 7.3.2 Coulombic electron gas and spin-polarized solutions
  16. 8 Conserving approximations: two-particle Green’s function
    1. 8.1 Introduction
    2. 8.2 Conditions on the approximate G2
    3. 8.3 Continuity equation
    4. 8.4 Momentum conservation law
    5. 8.5 Angular momentum conservation law
    6. 8.6 Energy conservation law
  17. 9 Conserving approximations: self-energy
    1. 9.1 Self-energy and Dyson equations I
    2. 9.2 Conditions on the approximate Σ
    3. 9.3 φ functional
    4. 9.4 Kadanoff–Baym equations
    5. 9.5 Fluctuation–dissipation theorem for the self-energy
    6. 9.6 Recovering equilibrium from the Kadanoff–Baym equations
    7. 9.7 Formal solution of the Kadanoff–Baym equations
  18. 10 MBPT for the Green’s function
    1. 10.1 Getting started with Feynman diagrams
    2. 10.2 Loop rule
    3. 10.3 Cancellation of disconnected diagrams
    4. 10.4 Summing only the topologically inequivalent diagrams
    5. 10.5 Self-energy and Dyson equations II
    6. 10.6 G-skeleton diagrams
    7. 10.7 W-skeleton diagrams
    8. 10.8 Summary and Feynman rules
  19. 11 MBPT and variational principles for the grand potential
    1. 11.1 Linked cluster theorem
    2. 11.2 Summing only the topologically inequivalent diagrams
    3. 11.3 How to construct the Φ functional
    4. 11.4 Dressed expansion of the grand potential
    5. 11.5 Luttinger–Ward and Klein functionals
    6. 11.6 Luttinger–Ward theorem
    7. 11.7 Relation between the reducible polarizability and the Φ functional
    8. 11.8 Ψ functional
    9. 11.9 Screened functionals
  20. 12 MBPT for the two-particle Green’s function
    1. 12.1 Diagrams for G2 and loop rule
    2. 12.2 Bethe–Salpeter equation
    3. 12.3 Excitons
    4. 12.4 Diagrammatic proof of K = ±δΣ/δG
    5. 12.5 Vertex function and Hedin equations
  21. 13 Applications of MBPT to equilibrium problems
    1. 13.1 Lifetimes and quasi-particles
    2. 13.2 Fluctuation–dissipation theorem for P and W
    3. 13.3 Correlations in the second-Born approximation
      1. 13.3.1 Polarization effects
    4. 13.4 Ground-state energy and correlation energy
    5. 13.5 GW correlation energy of a Coulombic electron gas
    6. 13.6 T-matrix approximation
      1. 13.6.1 Formation of a Cooper pair
  22. 14 Linear response theory: preliminaries
    1. 14.1 Introduction
    2. 14.2 Shortcomings of the linear response theory
      1. 14.2.1 Discrete–discrete coupling
      2. 14.2.2 Discrete–continuum coupling
      3. 14.2.3 Continuum–continuum coupling
    3. 14.3 Fermi golden rule
    4. 14.4 Kubo formula
  23. 15 Linear response theory: many-body formulation
    1. 15.1 Current and density response function
    2. 15.2 Lehmann representation
      1. 15.2.1 Analytic structure
      2. 15.2.2 The f-sum rule
      3. 15.2.3 Noninteracting fermions
    3. 15.3 Bethe–Salpeter equation from the variation of a conserving G
    4. 15.4 Ward identity and the f-sum rule
    5. 15.5 Time-dependent screening in an electron gas
      1. 15.5.1 Noninteracting density response function
      2. 15.5.2 RPA density response function
      3. 15.5.3 Sudden creation of a localized hole
      4. 15.5.4 Spectral properties in the G0W0 approximation
  24. 16 Applications of MBPT to nonequilibrium problems
    1. 16.1 Kadanoff–Baym equations for open systems
    2. 16.2 Time-dependent quantum transport: an exact solution
      1. 16.2.1 Landauer–Büttiker formula
    3. 16.3 Implementation of the Kadanoff–Baym equations
      1. 16.3.1 Time-stepping technique
      2. 16.3.2 Second-Born and GW self-energies
    4. 16.4 Initial-state and history dependence
    5. 16.5 Charge conservation
    6. 16.6 Time-dependent GW approximation in open systems
      1. 16.6.1 Keldysh Green’s functions in the double-time plane
      2. 16.6.2 Time-dependent current and spectral function
      3. 16.6.3 Screened interaction and physical interpretation
    7. 16.7 Inbedding technique: how to explore the reservoirs
    8. 16.8 Response functions from time-propagation
  25. A From the N roots of 1 to the Dirac δ-function
  26. B Graphical approach to permanents and determinants
  27. C Density matrices and probability interpretation
  28. D Thermodynamics and statistical mechanics
  29. E Green’s functions and lattice symmetry
  30. F Asymptotic expansions
  31. G Wick’s theorem for general initial states
  32. H BBGKY hierarchy
  33. I From δ-like peaks to continuous spectral functions
  34. J Virial theorem for conserving approximations
  35. K Momentum distribution and sharpness of the Fermi surface
  36. L Hedin equations from a generating functional
  37. M Lippmann–Schwinger equation and cross-section
  38. N Why the name Random Phase Approximation?
  39. O Kramers–Kronig relations
  40. P Algorithm for solving the Kadanoff–Baym equations
  41. References
  42. Index