This book provides a pedagogical introduction to the concepts and methods of quantum field theory necessary for the study of condensed matter and ultracold atomic gases. After a thorough discussion of the basic methods of field theory and many-body physics (functional integrals, perturbation theory, Feynman diagrams, correlation functions and linear response theory, symmetries and their consequences, etc.), the book covers a wide range of topics, from electron gas and Fermi-liquid theory to superfluidity and superconductivity, magnetic instabilities in electron systems, and dynamical mean-field theory of Mott transition. The focus is on the study of model Hamiltonians, where the microscopic physics and characteristic energy scales are encoded into a few effective parameters, rather than first-principle methods which start from a realistic Hamiltonian at the microscopic level and then make material-specific predictions. The reader is expected to be familiar with elementary quantum mechanics and statistical physics, and some acquaintance with condensed-matter physics and ultracold gases may also be useful. No prior knowledge of field theory or many-body problem is required.

Table of Contents

Part 1: Functional Integrals, Symmetries, and Correlation Functions

1. Functional Integrals

2. Symmetries

3. Correlation and Response Functions

Part 2: Quantum Many-Body Systems (I)

4. Fermi-Liquid Theory

5. The Electron Gas

6. Magnetism in Lattice Fermion Systems

7. Superfluidity and Superconductivity

8. Quantum Magnetism

9. Baym- Kadanoff Formalism