A course in quantum many-body theory : from conventional Fermi liquids to strongly correlated systems / / Michele Fabrizio

• Record Nr.: UNISA996499865103316
• Autore: Fabrizio Michele
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 9783031163050; 9783031163043
• Descrizione fisica: 1 online resource (350 pages)
• Collana: Graduate texts in physics
• Disciplina: 530.144
• Soggetti: Many-body problem; Quantum theory
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Second Quantization -- 1.1 Fock States and Space -- 1.2 Fermionic Operators -- 1.2.1 Fermi Fields -- 1.2.2 Second Quantisation of Multiparticle Operators -- 1.3 Bosonic Operators -- 1.3.1 Bose Fields and Multiparticle Operators -- 1.4 Canonical Transformations -- 1.4.1 Canonical Transformations with Charge Non-conserving Hamiltonians -- 1.4.2 Harmonic Oscillators -- 1.5 Application: Electrons in a Box -- 1.6 Application: Electron Lattice Models and Emergence of Magnetism -- 1.6.1 Hubbard Models -- 1.6.2 Mott Insulators and Heisenberg Models -- 1.7 Application: Spin-Wave Theory -- 1.7.1 Classical Ground State -- 1.8 Beyond the Classical Limit: The Spin-Wave Approximation -- 1.8.1 Hamiltonian of Quantum Fluctuations -- 1.8.2 Spin-Wave Dispersion and Goldstone Theorem -- 1.8.3 Validity of the Approximation and the Mermin-Wagner Theorem -- 1.8.4 Order from Disorder -- Problems -- 2 Linear Response Theory -- 2.1 Linear Response Functions -- 2.2 Kramers-Kronig Relations -- 2.2.1 Symmetries -- 2.3 Fluctuation-Dissipation Theorem -- 2.4 Spectral Representation -- 2.5 Power Dissipation -- 2.5.1 Absorption/Emission Processes -- 2.5.2 Thermodynamic Susceptibilities -- 2.6 Application: Linear Response to an Electromagnetic Field -- 2.6.1 Quantisation of the Electromagnetic Field -- 2.6.2 System's Sources for the Electromagnetic Field -- 2.6.3 ; Optical Constants -- 2.6.4 Linear Response in the Longitudinal Case -- 2.6.5 Linear Response in the Transverse Case -- 2.6.6 Power Dissipated by the Electromagnetic Field -- Problems -- 3 Hartree-Fock Approximation -- 3.1 Hartree-Fock Approximation for Fermions at Zero Temperature -- 3.1.1 Alternative Approach -- 3.2 Hartree-Fock Approximation for Fermions at Finite Temperature -- 3.2.1 Saddle Point Solution -- 3.3 Time-Dependent Hartree-Fock Approximation.; 3.3.1 Bosonization of the Low-Energy Particle-Hole Excitations -- 3.4 Application: Antiferromagnetism in the Half-Filled Hubbard Model -- 3.4.1 Spin-Wave Spectrum by Time-Dependent Hartree-Fock -- Problems -- 4 Feynman Diagram Technique -- 4.1 Preliminaries -- 4.1.1 Imaginary-Time Ordered Products -- 4.1.2 Matsubara Frequencies -- 4.1.3 Single-Particle Green's Functions -- 4.2 Perturbation Expansion in Imaginary Time -- 4.2.1 Wick's Theorem -- 4.3 Perturbation Theory for the Single-Particle Green's Function … -- 4.3.1 Diagram Technique in Momentum and Frequency Space -- 4.3.2 The Dyson Equation -- 4.3.3 Skeleton Diagrams -- 4.3.4 Physical Meaning of the Self-energy -- 4.3.5 Emergence of Quasiparticles -- 4.4 Other Kinds of Perturbations -- 4.4.1 Scalar Potential -- 4.4.2 Coupling to Bosonic Modes -- 4.5 Two-Particle Green's Functions and Correlation Functions -- 4.5.1 Diagrammatic Representation of the Two-Particle Green's Function -- 4.5.2 Correlation Functions -- 4.6 Coulomb Interaction and Proper and Improper Response Functions -- 4.7 Irreducible Vertices and the Bethe-Salpeter Equations -- 4.7.1 Particle-Hole Channel -- 4.7.2 Particle-Particle Channel -- 4.7.3 Self-energy and Irreducible Vertices -- 4.8 The Luttinger-Ward Functional -- 4.8.1 Thermodynamic Potential -- 4.9 Ward-Takahashi Identities -- 4.9.1 Ward-Takahashi Identity for the Heat Density -- 4.10 Conserving Approximation Schemes -- 4.10.1 Conserving Hartree-Fock Approximation -- 4.10.2 Conserving GW Approximation -- 4.11 Luttinger's Theorem -- 4.11.1 Validity Conditions for Luttinger's Theorem -- 4.11.2 Luttinger's Theorem in Presence of Quasiparticles and in Periodic Systems -- Problems -- 5 Landau's Fermi Liquid Theory -- 5.1 Emergence of Quasiparticles Reexamined -- 5.2 Manipulating the Bethe-Salpeter Equation -- 5.2.1 A Lengthy but Necessary Preliminary Calculation.; 5.2.2 Interaction Vertex and Density-Vertices -- 5.3 Linear Response Functions -- 5.3.1 Response Functions of Densities Associated to Conserved Quantities -- 5.4 Thermodynamic Susceptibilities -- 5.4.1 Charge Compressibility -- 5.4.2 Spin Susceptibility -- 5.4.3 Specific Heat -- 5.5 Current-Current Response Functions -- 5.5.1 Thermal Response -- 5.5.2 Coulomb Interaction -- 5.6 Mott Insulators with a Luttinger Surface -- 5.7 Luttinger's Theorem and Quasiparticle Distribution Function -- 5.7.1 Oshikawa's Topological Derivation of Luttinger's Theorem -- 5.8 Quasiparticle Hamiltonian and Landau-Boltzmann Transport Equation -- 5.8.1 Landau-Boltzmann Transport Equation for Quasiparticles -- 5.8.2 Transport Equation in Presence of an Electromagnetic Field -- 5.9 Application: Transport Coefficients with Rotational Symmetry -- 6 Brief Introduction to Luttinger Liquids -- 6.1 What Is Special in One Dimension? -- 6.2 Interacting Spinless Fermions -- 6.2.1 Bosonized Expression of the Non-interacting Hamiltonian -- 6.2.2 Bosonic Representation of the Fermi Fields -- 6.2.3 Operator Product Expansion -- 6.2.4 Non-interacting Green's Functions and Density-Density Response Functions -- 6.2.5 Interaction -- 6.2.6 Interacting Green's Functions and Correlation Functions -- 6.2.7 Umklapp Scattering -- 6.2.8 Behaviour Close to the K=1/2 Marginal Case -- 6.3 Spin-1/2 Heisenberg Chain -- 6.4 The One-Dimensional Hubbard Model -- 6.4.1 Luttinger Versus Fermi Liquids -- Problems -- ; 7 Kondo Effect and the Physics of the Anderson Impurity Model -- 7.1 Brief Introduction to Scattering Theory -- 7.1.1 General Analysis of the Phase-Shifts -- 7.2 The Anderson Impurity Model -- 7.2.1 Non Interacting Impurity -- 7.2.2 Hartree-Fock Approximation -- 7.3 From the Anderson Impurity Model to the Kondo Model -- 7.3.1 The Emergence of Logarithmic Singularities and the Kondo Temperature.; 7.3.2 Anderson's Poor Man's Scaling -- 7.4 Noziéres's Local Fermi Liquid Theory -- 7.4.1 Ward-Takahashi Identity -- 7.4.2 Luttinger's Theorem and Thermodynamic Susceptibilities -- Problems.