Many-Body Techniques in Condensed Matter Physics : Lecture Notes and Exercises for an Introductory Course

✍ Scribed by Jaime Merino; Alfredo Levy Yeyati

• Publisher: Springer Nature Switzerland
• Year: 2024
• Tongue: English
• Leaves: 219
• Category: Library

✦ Table of Contents

Preface
Acknowledgements
General Bibliography
Contents
1 Introduction to Many-Particle Physics in Condensed Matter
1.1 Quantum Field Theory and Statistical Mechanics
1.2 Second Quantization
1.3 Model of Metals
1.4 Hubbard Model
1.5 Heisenberg Model
1.6 Anderson Model of Magnetic Impurities in Metals
Part I Equilibrium Many-Body Techniques
2 Introduction to Green-Function Methods
2.1 Time Evolution
2.2 Time-Ordered or Causal Green's Functions at upper T equals 0T=0
2.3 Causal Green Function for Free Fermions
2.4 Physical Observables
2.5 General Analytic Properties of Green Functions in Interacting Systems
2.6 Retarded and Advanced Green Function
2.7 Fermi Liquid Properties
3 Perturbation Theory at Zero Temperature
3.1 Interaction Picture
3.2 Time-Evolution Operator
3.3 Relation Between Heisenberg and Interaction Pictures
3.4 Adiabatic Hypothesis
3.5 Gell-Mann and Low Theorem
3.6 Wick's Theorem
3.7 Diagrammatic Approach in Coordinate Space
3.8 Diagrammatic Approach in Momentum Space
3.9 Self-energies and Dyson Equation
3.10 Physical Interpretation of the Self-energy
4 Finite Temperature Green Function Formalism
4.1 Schrödinger Representation
4.2 Heisenberg Representation
4.3 Imaginary Time Green Function
4.3.1 Matsubara Representation of the Imaginary Time Green Function
4.4 Spectral Decomposition of Matsubara Green Function
4.5 Perturbation Theory at Finite Temperature
4.5.1 Interaction Representation
4.5.2 Perturbation Theory and Wick's Theorem
4.6 Feynman Rules for the Coulomb Interaction at Finite-upper TT
4.7 Feynman Diagrams at Finite Temperature
4.7.1 Self-consistent Hartree-Fock Approach
4.8 Diagrammatic Approach to the Electron-Phonon Interaction
4.8.1 Electron-Phonon Interaction in Second Quantization
4.8.2 Feynman Diagrams
4.9 Evaluation of Matsubara Sums
5 Linear Response and Collective Modes
5.1 Collective Modes in a Fermi Liquid
5.2 RPA Approximation
5.3 Screening of Coulomb Repulsion
5.4 Screening of an External Potential in the Electron Liquid
6 Spontaneous Symmetry Breaking and Mean-Field Theory
6.1 Generalized Green Function Propagator
6.2 Application to Metallic Ferromagnetism
6.3 Elementary Excitations in a Broken Symmetry Phase
Part II Non-equilibrium Many-Body Techniques
7 Introduction to Non-equilibrium: The Keldysh Contour
8 Perturbative Expansion in the Non-equilibrium Formalism
8.1 Basic Properties of Non-equilibrium GFs
8.2 Perturbative Expansion for a One-Body Perturbation
8.3 Case of Many-Body Interactions
8.4 The Dyson Equation as an Intregro-Differential Equation
8.5 The Triangular Representation
8.6 Langreth Rules
8.7 Frequency Representation
9 Applications: Electron Transport at the Nanoscale
9.1 Current Fluctuations
9.2 Full Counting Statistics
9.3 Superconducting Transport
9.3.1 Nambu Formalism
9.3.2 N/S Channel: Andreev Reflection
9.3.3 S/S Channel: Josephson Effect
9.3.4 Transport Through Majorana Nanowires
Part III Path Integral Formulation of the Quantum Many-Body Problem
10 Introduction to Path Integral Methods
10.1 Path Integral of a Single Boson
10.1.1 Green Function from Path Integration
10.1.2 Single Boson
10.1.3 Single Fermion
10.1.4 Wick's Theorem and Perturbation Theory
10.2 Path Integral Formulation for Non-equilibrium Systems
10.2.1 The Case of Lattice Models for Electron Transport
11 Application of Path Integral Methods: The Renormalization Group Approach
11.1 Non-interacting Fermion Model Under RG: Scale Invariance and Fixed Points
11.2 The One-Dimensional Hubbard Model Under RG: Spin-Charge Separation
11.2.1 RG Perturbative Treatment
11.2.2 First Order Corrections
11.2.3 Second Order Corrections
11.2.4 Spin Sector
11.2.5 Charge Sector
11.2.6 Discussion of RG Analysis of the 1D Hubbard Model
Appendix Hints for Solving Exercises
Appendix References