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Feynman Diagram Techniques in Condensed Matter Physics

Book Description

A concise introduction to Feynman diagram techniques, this book shows how they can be applied to the analysis of complex many-particle systems, and offers a review of the essential elements of quantum mechanics, solid state physics and statistical mechanics. Alongside a detailed account of the method of second quantization, the book covers topics such as Green's and correlation functions, diagrammatic techniques and superconductivity, and contains several case studies. Some background knowledge in quantum mechanics, solid state physics and mathematical methods of physics is assumed. Detailed derivations of formulas and in-depth examples and chapter exercises from various areas of condensed matter physics make this a valuable resource for both researchers and advanced undergraduate students in condensed matter theory, many-body physics and electrical engineering. Solutions to exercises are available online.

Table of Contents

  1. Cover
  2. Half Title
  3. Title
  4. Copyright
  5. Dedication
  6. Table of Contents
  7. Preface
  8. 1 A brief review of quantum mechanics
    1. 1.1 The postulates
    2. 1.2 The harmonic oscillator
    3. Further reading
    4. Problems
  9. 2 Single-particle states
    1. 2.1 Introduction
    2. 2.2 Electron gas
    3. 2.3 Bloch states
    4. 2.4 Example: one-dimensional lattice
    5. 2.5 Wannier states
    6. 2.6 Two-dimensional electron gas in a magnetic field
    7. Further reading
    8. Problems
  10. 3 Second quantization
    1. 3.1 N-particle wave function
    2. 3.2 Properly symmetrized products as a basis set
    3. 3.3 Three examples
    4. 3.4 Creation and annihilation operators
      1. 3.4.1 Fermions
      2. 3.4.2 Bosons
    5. 3.5 One-body operators
    6. 3.6 Examples
      1. 3.6.1 Kinetic energy of a system of N electrons
      2. 3.6.2 External potential
      3. 3.6.3 Particle-number density
    7. 3.7 Two-body operators
    8. 3.8 Translationally invariant system
    9. 3.9 Example: Coulomb interaction
    10. 3.10 Electrons in a periodic potential
      1. 3.10.1 Bloch representation
      2. 3.10.2 Wannier representation
    11. 3.11 Field operators
      1. 3.11.1 Definition
      2. 3.11.2 Commutation relations
      3. 3.11.3 One-body operators
      4. 3.11.4 Two-body operators
      5. 3.11.5 Examples
        1. 3.11.5.1 Particle-number density
        2. 3.11.5.2 Kinetic energy
    12. Further reading
    13. Problems
  11. 4 The electron gas
    1. 4.1 The Hamiltonian in the jellium model
    2. 4.2 High density limit
    3. 4.3 Ground state energy
      1. 4.3.1 First order perturbation
      2. 4.3.2 Second order perturbation
    4. Further reading
    5. Problems
  12. 5 A brief review of statistical mechanics
    1. 5.1 The fundamental postulate of statistical mechanics
    2. 5.2 Contact between statistics and thermodynamics
    3. 5.3 Ensembles
      1. 5.3.1 The microcanonical ensemble
      2. 5.3.2 The canonical ensemble
      3. 5.3.3 The grand canonical ensemble
    4. 5.4 The statistical operator for a general ensemble
      1. 5.4.1 Definition
      2. 5.4.2 General properties
      3. 5.4.3 Time evolution
    5. 5.5 Quantum distribution functions
    6. Further reading
    7. Problems
  13. 6 Real-time Green’s and correlation functions
    1. 6.1 A plethora of functions
      1. 6.1.1 Correlation functions
      2. 6.1.2 Time dependence
      3. 6.1.3 Single-particle Green’s functions
    2. 6.2 Physical meaning of Green’s functions
    3. 6.3 Spin-independent Hamiltonian, translational invariance
    4. 6.4 Spectral representation
      1. 6.4.1 Retarded and advanced Green’s functions
      2. 6.4.2 Single-particle correlation function
      3. 6.4.3 Retarded correlation function
      4. 6.4.4 Correlation function
    5. 6.5 Example: Green’s function of a noninteracting system
      1. 6.5.1 Derivation from the spectral density function
      2. 6.5.2 An alternative derivation
    6. 6.6 Linear response theory
    7. 6.7 Noninteracting electron gas in an external potential
    8. 6.8 Dielectric function of a noninteracting electron gas
    9. 6.9 Paramagnetic susceptibility of a noninteracting electron gas
    10. 6.10 Equation of motion
    11. 6.11 Example: noninteracting electron gas
    12. 6.12 Example: an atom adsorbed on graphene
    13. Further reading
    14. Problems
  14. 7 Applications of real-time Green’s functions
    1. 7.1 Single-level quantum dot
    2. 7.2 Quantum dot in contact with a metal: Anderson’s model
    3. 7.3 Tunneling in solids
    4. Further reading
    5. Problems
  15. 8 Imaginary-time Green’s and correlation functions
    1. 8.1 Imaginary-time correlation function
      1. 8.1.1 Time-dependence
      2. 8.1.2 Periodicity
    2. 8.2 Imaginary-time Green’s function
    3. 8.3 Significance of the imaginary-time Green’s function
    4. 8.4 Spectral representation, relation to real-time functions
      1. 8.4.1 Imaginary-time Green’s function
      2. 8.4.2 Imaginary-time correlation function
    5. 8.5 Example: Green’s function for noninteracting particles
      1. 8.5.1 Derivation from the spectral density function
      2. 8.5.2 An alternative derivation
    6. 8.6 Example: Green’s function for 2-DEG in a magnetic field
    7. 8.7 Green’s function and the U-operator
      1. 8.7.1 The Interaction picture
      2. 8.7.2 The U-operator
      3. 8.7.3 Green’s function and the U-operator
      4. 8.7.4 Perturbation expansion of the imaginary-time Green’s function
    8. 8.8 Wick’s theorem
      1. 8.8.1 Contractions
      2. 8.8.2 Statement of Wick’s theorem
      3. 8.8.3 An example
      4. 8.8.4 Some useful results
      5. 8.8.5 Proof of Wick’s theorem
      6. 8.8.6 Some remarks on Wick’s theorem
    9. 8.9 Case study: first-order interaction
    10. 8.10 Cancellation of disconnected diagrams
    11. Further reading
    12. Problems
  16. 9 Diagrammatic techniques
    1. 9.1 Case study: second-order perturbation in a system of fermions
    2. 9.2 Feynman rules in momentum-frequency space
    3. 9.3 An example of how to apply Feynman rules
    4. 9.4 Feynman rules in coordinate space
    5. 9.5 Self energy and Dyson’s equation
    6. 9.6 Energy shift and the lifetime of excitations
    7. 9.7 Time-ordered diagrams: a case study
    8. 9.8 Time-ordered diagrams: Dzyaloshinski’s rules
    9. Further reading
    10. Problems
  17. 10 Electron gas: a diagrammatic approach
    1. 10.1 Model Hamiltonian
    2. 10.2 The need to go beyond first-order perturbation theory
    3. 10.3 Second-order perturbation theory: still inadequate
    4. 10.4 Classification of diagrams according to the degree of divergence
    5. 10.5 Self energy in the random phase approximation (RPA)
    6. 10.6 Summation of the ring diagrams
    7. 10.7 Screened Coulomb interaction
    8. 10.8 Collective electronic density fluctuations
    9. 10.9 How do electrons interact?
    10. 10.10 Dielectric function
      1. 10.10.1 Thomas–Fermi screening model
    11. 10.11 Plasmons and Landau damping
      1. 10.11.1 plasmons
      2. 10.11.2 Landau damping
    12. 10.12 Case study: dielectric function of graphene
    13. Further reading
    14. Problems
  18. 11 Phonons, photons, and electrons
    1. 11.1 Lattice vibrations in one dimension
    2. 11.2 One-dimensional diatomic lattice
    3. 11.3 Phonons in three-dimensional crystals
    4. 11.4 Phonon statistics
    5. 11.5 Electron–phonon interaction: rigid-ion approximation
    6. 11.6 Electron–LO phonon interaction in polar crystals
    7. 11.7 Phonon Green’s function
      1. 11.7.1 Definitions
      2. 11.7.2 Periodicity
    8. 11.8 Free-phonon Green’s function
    9. 11.9 Feynman rules for the electron–phonon interaction
    10. 11.10 Electron self energy
    11. 11.11 The electromagnetic field
    12. 11.12 Electron–photon interaction
    13. 11.13 Light scattering by crystals
    14. 11.14 Raman scattering in insulators
    15. Further reading
    16. Problems
  19. 12 Superconductivity
    1. 12.1 Properties of superconductors
    2. 12.2 The London equation
    3. 12.3 Effective electron–electron interaction
    4. 12.4 Cooper pairs
    5. 12.5 BCS theory of superconductivity
    6. 12.6 Mean field approach
    7. 12.7 Green’s function approach to superconductivity
    8. 12.8 Determination of the transition temperature
    9. 12.9 The Nambu formalism
    10. 12.10 Response to a weak magnetic field
    11. 12.11 Infinite conductivity
    12. Further reading
    13. Problems
  20. 13 Nonequilibrium Green’s function
    1. 13.1 Introduction
    2. 13.2 Schrödinger, Heisenberg, and interaction pictures
      1. 13.2.1 The Schrödinger picture
      2. 13.2.2 The Heisenberg picture
      3. 13.2.3 The interaction picture
    3. 13.3 The malady and the remedy
    4. 13.4 Contour-ordered Green’s function
    5. 13.5 Kadanoff–Baym and Keldysh contours
    6. 13.6 Dyson’s equation
    7. 13.7 Langreth rules
    8. 13.8 Keldysh equations
    9. 13.9 Steady-state transport
      1. 13.9.1 Model Hamiltonian
      2. 13.9.2 Expression for the current
    10. 13.10 Noninteracting quantum dot
    11. 13.11 Coulomb blockade in the Anderson model
    12. Further reading
    13. Problems
  21. Appendix A Second quantized form of operators
    1. A.1 Fermions
      1. A.1.1 One-body operators
      2. A.1.2 Two-body operators
    2. A.2 Bosons
      1. A.2.1 One-body operators
      2. A.2.2 Two-body operators
  22. Appendix B Completing the proof of Dzyaloshinski’s rules
  23. Appendix C Lattice vibrations in three dimensions
    1. C.1 Harmonic approximation
    2. C.2 Classical theory of lattice vibrations
    3. C.3 Vibrational energy
    4. C.4 Quantum theory of lattice vibrations
  24. Appendix D Electron–phonon interaction in polar crystals
    1. D.1 Polarization
    2. D.2 Electron–LO phonon interaction
  25. References
  26. Index