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Feynman diagram techniques in condensed matter physics / Radi A. Jishi, California State University.

By: Jishi, Radi A, 1955-Material type: TextTextPublication details: Cambridge : Cambridge University Press, 2013. Description: 1 online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9781107341753; 1107341752; 113917777X; 9781139177771; 9781299707757; 1299707750; 9781107345508; 1107345502; 9781107357624; 1107357624; 9781107348004; 1107348005; 9781107655331; 1107655331Subject(s): Feynman diagrams | Many-body problem | Condensed matter | SCIENCE -- Physics | SCIENCE -- Physics -- Condensed Matter | Condensed matter | Feynman diagrams | Many-body problem | Kondensierte Materie | Feynman-GraphGenre/Form: Electronic book. | Electronic books. Additional physical formats: Print version:: Feynman diagram techniques in condensed matter physics.DDC classification: 530.4/1 LOC classification: QC794.6.F4 | J57 2013ebOther classification: SCI055000 Online resources: Click here to access online
Contents:
Preface -- 1 A brief review of quantum mechanics -- 1.1 The postulates -- (I) The quantum state -- (II) Observables -- (III) Time evolution -- (IV) Measurements -- (V) Wave function of a system of identical particles -- 1.2 The harmonic oscillator -- Further reading -- Problems -- 2 Single-particle states -- 2.1 Introduction -- 2.2 Electron gas -- 2.3 Bloch states -- 2.4 Example: one-dimensional lattice -- 2.5 Wannier states -- 2.6 Two-dimensional electron gas in a magnetic field -- Further reading -- Problems -- 3 Second quantization -- 3.1 N-particle wave function -- 3.2 Properly symmetrized products as a basis set -- 3.3 Three examples -- 3.4 Creation and annihilation operators -- 3.5 One-body operators -- 3.6 Examples -- 3.7 Two-body operators -- 3.8 Translationally invariant system -- 3.9 Example: Coulomb interaction -- 3.10 Electrons in a periodic potential -- 3.11 Field operators -- Further reading -- Problems -- 4 The electron gas -- 4.1 The Hamiltonian in the jellium model -- 4.2 High density limit -- 4.3 Ground state energy -- Further reading -- Problems -- 5 A brief review of statistical mechanics -- 5.1 The fundamental postulate of statistical mechanics -- 5.2 Contact between statistics and thermodynamics -- 5.3 Ensembles -- 5.4 The statistical operator for a general ensemble -- 5.5 Quantum distribution functions -- Further reading -- Problems -- 6 Real-time Green's and correlation functions -- 6.1 A plethora of functions -- 6.2 Physical meaning of Green's functions -- 6.3 Spin-independent Hamiltonian, translational invariance -- 6.4 Spectral representation -- 6.5 Example: Green's function of a noninteracting system -- 6.6 Linear response theory -- 6.7 Noninteracting electron gas in an external potential -- 6.8 Dielectric function of a noninteracting electron gas.
6.9 Paramagnetic susceptibility of a noninteracting electron gas -- 6.10 Equation of motion -- 6.11 Example: noninteracting electron gas -- 6.12 Example: an atom adsorbed on graphene -- Further reading -- Problems -- 7 Applications of real-time Green's functions -- 7.1 Single-level quantum dot -- 7.2 Quantum dot in contact with a metal: Anderson's model -- 7.3 Tunneling in solids -- Further reading -- Problems -- 8 Imaginary-time Green's and correlation functions -- 8.1 Imaginary-time correlation function -- 8.2 Imaginary-time Green's function -- 8.3 Significance of the imaginary-time Green's function -- 8.4 Spectral representation, relation to real-time functions -- 8.5 Example: Green's function for noninteracting particles -- 8.6 Example: Green's function for 2-DEG in a magnetic field -- 8.7 Green's function and the U-operator -- 8.8 Wick's theorem -- 8.9 Case study: first-order interaction -- 8.10 Cancellation of disconnected diagrams -- Further reading -- Problems -- 9 Diagrammatic techniques -- 9.1 Case study: second-order perturbation in a system of fermions -- 9.2 Feynman rules in momentum-frequency space -- 9.3 An example of how to apply Feynman rules -- 9.4 Feynman rules in coordinate space -- 9.5 Self energy and Dyson's equation -- 9.6 Energy shift and the lifetime of excitations -- 9.7 Time-ordered diagrams: a case study -- 9.8 Time-ordered diagrams: Dzyaloshinski's rules -- Further reading -- Problems -- 10 Electron gas: a diagrammatic approach -- 10.1 Model Hamiltonian -- 10.2 The need to go beyond first-order perturbation theory -- 10.3 Second-order perturbation theory: still inadequate -- 10.4 Classification of diagrams according to the degree of divergence -- 10.5 Self energy in the random phase approximation (RPA) -- 10.6 Summation of the ring diagrams -- 10.7 Screened Coulomb interaction.
10.8 Collective electronic density fluctuations -- 10.9 How do electrons interact? -- 10.10 Dielectric function -- 10.11 Plasmons and Landau damping -- 10.12 Case study: dielectric function of graphene -- Further reading -- Problems -- 11 Phonons, photons, and electrons -- 11.1 Lattice vibrations in one dimension -- 11.2 One-dimensional diatomic lattice -- 11.3 Phonons in three-dimensional crystals -- 11.4 Phonon statistics -- 11.5 Electron-phonon interaction: rigid-ion approximation -- 11.6 Electron-LO phonon interaction in polar crystals -- 11.7 Phonon Green's function -- 11.8 Free-phonon Green's function -- 11.9 Feynman rules for the electron-phonon interaction -- 11.10 Electron self energy -- 11.11 The electromagnetic field -- 11.12 Electron-photon interaction -- 11.13 Light scattering by crystals -- 11.14 Raman scattering in insulators -- Further reading -- Problems -- 12 Superconductivity -- 12.1 Properties of superconductors -- 12.2 The London equation -- 12.3 Effective electron-electron interaction -- 12.4 Cooper pairs -- 12.5 BCS theory of superconductivity -- 12.6 Mean field approach -- 12.7 Green's function approach to superconductivity -- 12.8 Determination of the transition temperature -- 12.9 The Nambu formalism -- 12.10 Response to a weak magnetic field -- 12.11 Infinite conductivity -- Further reading -- Problems -- 13 Nonequilibrium Green's function -- 13.1 Introduction -- 13.2 Schrodinger, Heisenberg, and interaction pictures -- 13.3 The malady and the remedy -- 13.4 Contour-ordered Green's function -- 13.5 Kadanoff-Baym and Keldysh contours -- 13.6 Dyson's equation -- 13.7 Langreth rules -- 13.8 Keldysh equations -- 13.9 Steady-state transport -- 13.10 Noninteracting quantum dot -- 13.11 Coulomb blockade in the Anderson model -- Further reading -- Problems -- Appendix A: Second quantized form of operators -- A.1 Fermions.
A.2 Bosons -- Appendix B: Completing the proof of Dzyaloshinski's rules -- Appendix C: Lattice vibrations in three dimensions -- C.1 Harmonic approximation -- C.2 Classical theory of lattice vibrations -- C.3 Vibrational energy -- C.4 Quantum theory of lattice vibrations -- Appendix D: Electron-phonon interaction in polar crystals -- D.1 Polarization -- D.2 Electron-LO phonon interaction -- References -- Index.
Summary: "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, 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 made available online"-- Provided by publisher.
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"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, 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 made available online"-- Provided by publisher.

Includes bibliographical references and index.

Print version record.

Preface -- 1 A brief review of quantum mechanics -- 1.1 The postulates -- (I) The quantum state -- (II) Observables -- (III) Time evolution -- (IV) Measurements -- (V) Wave function of a system of identical particles -- 1.2 The harmonic oscillator -- Further reading -- Problems -- 2 Single-particle states -- 2.1 Introduction -- 2.2 Electron gas -- 2.3 Bloch states -- 2.4 Example: one-dimensional lattice -- 2.5 Wannier states -- 2.6 Two-dimensional electron gas in a magnetic field -- Further reading -- Problems -- 3 Second quantization -- 3.1 N-particle wave function -- 3.2 Properly symmetrized products as a basis set -- 3.3 Three examples -- 3.4 Creation and annihilation operators -- 3.5 One-body operators -- 3.6 Examples -- 3.7 Two-body operators -- 3.8 Translationally invariant system -- 3.9 Example: Coulomb interaction -- 3.10 Electrons in a periodic potential -- 3.11 Field operators -- Further reading -- Problems -- 4 The electron gas -- 4.1 The Hamiltonian in the jellium model -- 4.2 High density limit -- 4.3 Ground state energy -- Further reading -- Problems -- 5 A brief review of statistical mechanics -- 5.1 The fundamental postulate of statistical mechanics -- 5.2 Contact between statistics and thermodynamics -- 5.3 Ensembles -- 5.4 The statistical operator for a general ensemble -- 5.5 Quantum distribution functions -- Further reading -- Problems -- 6 Real-time Green's and correlation functions -- 6.1 A plethora of functions -- 6.2 Physical meaning of Green's functions -- 6.3 Spin-independent Hamiltonian, translational invariance -- 6.4 Spectral representation -- 6.5 Example: Green's function of a noninteracting system -- 6.6 Linear response theory -- 6.7 Noninteracting electron gas in an external potential -- 6.8 Dielectric function of a noninteracting electron gas.

6.9 Paramagnetic susceptibility of a noninteracting electron gas -- 6.10 Equation of motion -- 6.11 Example: noninteracting electron gas -- 6.12 Example: an atom adsorbed on graphene -- Further reading -- Problems -- 7 Applications of real-time Green's functions -- 7.1 Single-level quantum dot -- 7.2 Quantum dot in contact with a metal: Anderson's model -- 7.3 Tunneling in solids -- Further reading -- Problems -- 8 Imaginary-time Green's and correlation functions -- 8.1 Imaginary-time correlation function -- 8.2 Imaginary-time Green's function -- 8.3 Significance of the imaginary-time Green's function -- 8.4 Spectral representation, relation to real-time functions -- 8.5 Example: Green's function for noninteracting particles -- 8.6 Example: Green's function for 2-DEG in a magnetic field -- 8.7 Green's function and the U-operator -- 8.8 Wick's theorem -- 8.9 Case study: first-order interaction -- 8.10 Cancellation of disconnected diagrams -- Further reading -- Problems -- 9 Diagrammatic techniques -- 9.1 Case study: second-order perturbation in a system of fermions -- 9.2 Feynman rules in momentum-frequency space -- 9.3 An example of how to apply Feynman rules -- 9.4 Feynman rules in coordinate space -- 9.5 Self energy and Dyson's equation -- 9.6 Energy shift and the lifetime of excitations -- 9.7 Time-ordered diagrams: a case study -- 9.8 Time-ordered diagrams: Dzyaloshinski's rules -- Further reading -- Problems -- 10 Electron gas: a diagrammatic approach -- 10.1 Model Hamiltonian -- 10.2 The need to go beyond first-order perturbation theory -- 10.3 Second-order perturbation theory: still inadequate -- 10.4 Classification of diagrams according to the degree of divergence -- 10.5 Self energy in the random phase approximation (RPA) -- 10.6 Summation of the ring diagrams -- 10.7 Screened Coulomb interaction.

10.8 Collective electronic density fluctuations -- 10.9 How do electrons interact? -- 10.10 Dielectric function -- 10.11 Plasmons and Landau damping -- 10.12 Case study: dielectric function of graphene -- Further reading -- Problems -- 11 Phonons, photons, and electrons -- 11.1 Lattice vibrations in one dimension -- 11.2 One-dimensional diatomic lattice -- 11.3 Phonons in three-dimensional crystals -- 11.4 Phonon statistics -- 11.5 Electron-phonon interaction: rigid-ion approximation -- 11.6 Electron-LO phonon interaction in polar crystals -- 11.7 Phonon Green's function -- 11.8 Free-phonon Green's function -- 11.9 Feynman rules for the electron-phonon interaction -- 11.10 Electron self energy -- 11.11 The electromagnetic field -- 11.12 Electron-photon interaction -- 11.13 Light scattering by crystals -- 11.14 Raman scattering in insulators -- Further reading -- Problems -- 12 Superconductivity -- 12.1 Properties of superconductors -- 12.2 The London equation -- 12.3 Effective electron-electron interaction -- 12.4 Cooper pairs -- 12.5 BCS theory of superconductivity -- 12.6 Mean field approach -- 12.7 Green's function approach to superconductivity -- 12.8 Determination of the transition temperature -- 12.9 The Nambu formalism -- 12.10 Response to a weak magnetic field -- 12.11 Infinite conductivity -- Further reading -- Problems -- 13 Nonequilibrium Green's function -- 13.1 Introduction -- 13.2 Schrodinger, Heisenberg, and interaction pictures -- 13.3 The malady and the remedy -- 13.4 Contour-ordered Green's function -- 13.5 Kadanoff-Baym and Keldysh contours -- 13.6 Dyson's equation -- 13.7 Langreth rules -- 13.8 Keldysh equations -- 13.9 Steady-state transport -- 13.10 Noninteracting quantum dot -- 13.11 Coulomb blockade in the Anderson model -- Further reading -- Problems -- Appendix A: Second quantized form of operators -- A.1 Fermions.

A.2 Bosons -- Appendix B: Completing the proof of Dzyaloshinski's rules -- Appendix C: Lattice vibrations in three dimensions -- C.1 Harmonic approximation -- C.2 Classical theory of lattice vibrations -- C.3 Vibrational energy -- C.4 Quantum theory of lattice vibrations -- Appendix D: Electron-phonon interaction in polar crystals -- D.1 Polarization -- D.2 Electron-LO phonon interaction -- References -- Index.

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