Presents functional integral methods of quantum many-body theory. Starting with Feynman’s path integral, it develops functional integrals of partition functions in imaginary time and extends these techniques to many-body systems. It expands the formulation in the coherent-state basis, and describes the application of the Hubbard–Stratononvich transformation and the saddle-point approximation.

Deals with magnetism in itinerant systems. It starts with the Stoner mean-field theory, as derived from a simple Hubbard model, and Stoner excitations and spin-waves obtained with the aid of RPA. The concept of nesting and spin-density waves is then discussed. This is followed by a detailed presentation of Anderson’s magnetic impurity model and its relation to the Kondo model through the Schrieffer–Wolff transformation. Finally, a detailed account of the Kondo effect and the Kondo resonance is given.

Covers linear response from the one-electron viewpoint, including causality and the Kramers–Kronig relation. It develops the Kubo conductivity formula with special reference to the quantum Hall effect. The longitudinal and transverse dielectric functions are derived, and the ideas of intraband and interband, both direct and indirect, optical transitions are discussed.

Develops the one-particle formalism within Hartree–Fock and density functional frameworks,and examines validity bounds. The effects of exchange and correlation are also discussed, bringing out the idea of an exchange hole for fermions.

Explains the effect of dimensionality on electronic susceptibilities, including nesting effects. It describes the onset of instabilities, as manifest in the Peierls phenomenon, and delineates their emergent order parameters. It also introduces the idea of a Kohn anomaly, and derives the giant Kohn anomaly as a consequence of the one-dimensional Peierls instability.

Introduces the concept of Cooper pairing and develops a diagramatic approach to the Cooper instability. The BCS Hamiltonian is then constructed and solved with the aid of the Bogoliubov–Valatin transformation. Nambu–Gorkov formalism is introduced, and the Gor'kov anomalous Green function constructed. The Ginzburg–Landau formulation is derived from microscopic theory using the coherent-state partition function and the HS transformation. Detailed account of the Ginzburg–Landau perspective of superconductivity is given, ending with a derivation of the Meissner effect and an explanation of the Anderson–Higgs mechanism.

Reveals limitations of noninteracting fermion formulation. The chapter also introduces Landau’s Fermi liquid parameters and the conceptual basis of quasiparticles. Some suscptibilities are derived. Microscopic justification is explained.

Treats the Bose-Einstein condensation, and explains superfluidity from the Bogoliubov and Ginzburg-Landau perspectives. It also describes the concept of spontaneous symmetry breaking and Goldstone modes.

Develops the formalism of electron–phonon interaction within the Matsubara framework.

Presents relevant aspects of topology, such as homeomorphism, fiber and vector bundles, connection and curvature, parallel transport, and holonomy, and ends with establishing the relevance of topology to physics.

Dirac materials and Dirac fermions are presented. Graphene, with its Dirac points and cones, and the Dirac fermion Hamiltonian in the vicinity of the K-points are described. Time-reversal symmetry-breaking Chern insulators, with special focus on Haldane’s model, are presented. The quantum spin Hall effect, as manifest in the graphene-like model of Kane–Mele, with strong spin–orbit (SO) coupling, is outlined. A detailed description of Weyl semimetals is given.

Outlines the different methods of electronic band calculations with detailed presentation of the pseudopotential and tight-binding methods, including Harrison’s matrix element scaling.

Presents a Hartree–Fock perturbative treatment of the interacting electron gas within the jellium model and highlights its drawbacks. It also introduces the concept of the random phase approximation.

Covers ferromagnetic and antiferromagnetic insulators, describing the nature of their respective ground states and deriving their spin-wave excitation spectra with the aid of the Holstein–Primakoff transformation.