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.

Develops the many-body one-particle Green function, and explains its physical interpretation in terms of the spectral function, self-energy, and quasiparticle. lifetime. Its application in angle-resolved photoemission spectroscopy is presented in detail. The time-evolution operator in the interaction picture is derived, and time-ordering and adiabatic switching-on are introduced as precursor tools to construct the Feynman–Dyson many-body perturbation theory. A detailed account of Wick’s theorem, normal ordering, and contractions is outlined. Feynman diagrams are constructed, and the emergence of the infinite Dyson series from irreducible diagrams is outlined. Two-particle Green function and the particle-hole excitation spectra are developed. Diagramatic application of RPA for interacting systems is described. The finite-temperature Matsubara Green function is introduced and developed, together with its Fourier series representation and the evaluation of Matsubara sums.

Presents a detailed account of phonons, lattice dynamics and experimental techniques for measuring phonon dispersions. It derives the electron-phonon coupling in terms of symmetry-adapted (or normal) mode coordinates. The electronic contributions to phonon energies are developed in terms of the density-density response function. The developed expressions are then pedagogically used to construct phenomenological models for phonons in semiconductors and insulators "pseudocharge models”. The different experimental probes used in measuring the static and dynamic structure factors of solids are introduced.

Describes Berry’s phase, connection, and curvature, and derives the Chern topological number. It presents two pedagogically important but distinct examples: a two-level system, with its concomitant “magnetic monopole," and the molecular Aharonov–Bohm effect, where the interplay between the quantum electronic andionic motions leads to fascinating topological manifestations.

Presents a short review of the historical development of topology in condensed matter physics.

Treats non-Fermi liquids and quantum critical points and describes Luttinger liquid theories. Bosonization of the noninteracting and interacting Tomonaga–Luttinger models are derived. Bosonization of the single particle operators is establshed and the corresponding Green functions derived.

The chapter presents a theory of electron transport in graphene and discussion of the corresponding experimental data. We start with the discussion of quantum and classical Boltzmann equations and Kubo–Nakano–Moti formula for the electric resistivity. Further, we discuss the main extrinsic scattering mechanicsms relevant for the transport (charge impurities, resonant impurities, static ripples), and intrinsic mobility. For the latter, the role of two-phonon processes invloving flexural phonons is especially emphasized. We also consider edge scattering in graphene nanoribbons. Further, we discuss nonlocal electron transport, weak localization effects, and hydrodynamics of electron liquid in graphene.

This chapter mostly explains the role of graphene as a prototype crystalline membrane. We discuss peculiarities of phonon spectra of two-dimensional crystals, such as existence of soft flexural modes and unavoidably decisive role of anharmonic effects, the physical origin of negative thermal expansion of graphene and Mermin–Wagner theorem forbidding long-range crystalline order for two-dimensional materials. We consider mechanics and statistical mechanics of crystalline membranes and especially the role of thermal fluctuations resulting in intrinsic ripples. At the end of this chapter, we give a basic introduction to Raman spectroscopy which is one of the most important experimental tools to probe the properties of graphene.