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.

Starting from a detailed explanation of Klein paradox of relativistic quantum mechanics, we consider a motion of massless Dirac fermions through potential barriers. It is shown that chiral properties of these particles guarantee a penetration through arbitrarily high and broad potential barriers. The role of this phenomenon (chiral tunneling) for graphene physics and technology is explained. We discuss analogy between electronic optics of graphene and optical properties of metamaterials, especially, Veselago lensing effect for massless Dirac fermions. Chiral tunneling in bilayer graphene is discussed.

Using graphene on hexagonal boron nitride (hBN) as an example, we introduce the concept of van der Waals heterostructures. First, we explain extraordinary high quality of graphene on hBN. Then we discuss the physics of formation of moiré patterns and a general problem of commensurability and incommensurability. We also discuss the basic consequences for electronic structure and electron transport properties, including a conductivity along zero-mass lines, formation of additional Dirac points and recently experimentally discovered new types of magneto-oscillation effects in graphene superlattuces.

We discuss the physics of pseudomagnetic field,s which can be induced in graphene by applying strains, and show how they can be used to manipulate electronic transport through graphene heterostructures (strain engineering). We consider strain-induced pseudo-Landau levels, which were observed in graphene, and discuss the related valley quantum Hall effect. At the end of this chapter we demonstrate that a combination of strain and electric gating can open energy gap in electron spectrum of graphene which can be potentially interesting for applications.

Exact solution of two-dimensional Dirac equation for Coulomb potential (Dirac–Kepler problem) is presented. Linear and nonlinear screening of the Coulomb potential is discused. The main focus is on the phenomenon of relativistic collapse of supercritical charges, which was discussed for many years in high-energy physics (this is the process that determines the end of the periodic table) and was at last discovered in graphene. We introduce Hartree–Fock theory for massless Dirac electrons and show that their Coulomb interaction essentially renormalizes Fermi velocity in such a way that Dirac cone is, strictly speaking, no more cone.

We continue the discussion of Van der Waals heterostructures for the case of twisted blayer graphene. After a general consideration, we discuss a special case of graphene dodecagonal quasicrystal for misorientation angle 30⁰. We also discuss a formation of flat bands for small misorientation angles and give a brief introduction to the physics of flat electron bands.

The chapter explains a physics of minimal conductivity in graphene. It is shown that a new type of electronic transport arises in graphene, namely, electron tunneling via zero modes of Dirac operator. The relation to Zitterbewegung concept of relativistic quantum mechanics is demonstrated. We calculate the value of minuimal conductivity and shot noise in the neutrality point, and consider Aharonov–Bohm effect in undoped graphene rings.

Optical properties of massless Dirac electrons are considered. In particular, it is shown that they provide a universal, frequency-independent adsorption coefficient determined by fine structure constant. The possible effect of interelectron interaction on this property is discussed. Using a perturbation theory for density matrix, we derive Kubo formula for various response functions and use it to consider optics, magnetooptics, charge screening and diamagnetism of massless Dirac electrons. Graphene plasmonics is briefly reviewed.