A Course in Quantum Many-Body Theory

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