The WKB method is the resolution method of choice for singularly perturbed linear ordinary differential equations. It is introduced by the historical example of its first user, Francesco Carlini, in his treatment of Kepler’s two-body problem in 1817. At that stage, the presentation of the method is not yet complete, but one is already capable of solving boundary layer problems and wave and vibrations problems and discussing Berry’s phase in quantum mechanics. Then, the presence of singularities, known as turning points, is explained and treated in a number of applications ranging from quantum mechanics to fluid instabilities. On either side of a turning point, some constants of integration change value, a mathematical behaviour called the Stokes phenomenon. We discuss it in detail and relate this effect to the divergence of the asymptotic series generated by the WKB method. This prepares the reader for Chapter 7 of the book. Second-order turning points and an alternative implementation of the method based on the Airy function are also presented.

A rich vein of physics and useful magnetic phenomena is associated with transverse magneto-transport of electron and spin currents. First is the spin Hall effect and the related anomalous Hall effect. Other variants arise as a result of spin-orbit coupling, chirality and the lack of a centre of symmetry in a crystal or thin film stack. Quantized electrical conductance is related to the Landau levels. Associated magnetic phenomena are Landau diamagnetism, quantized conductance and the quantum Hall effects in two-dimensional electronic systems. Dissipationless edge and surface currents arise that may be unpolarized as in the quantum Hall effect, or spin polarized, as in topological insulators. Possible ‘emergent’ magnetic fields are discussed and related to real space and reciprocal space lattice topology. Analogous effects associated with heat currents include the spin and anomalous Nernst effects.

This chapter offers a thorough understanding of polarization and its applications. We explore its historical development from static charge theories to modern insights involving Berry phases. Additionally, we examine ferroic materials.

When a quantum system has some external time dependence, some rather special things happen. This chapter explores this subject. Among the topics that we cover are the adiabatic theorem, Berry phase, the sudden approximation, and time-dependent perturbation theory.

The chapter starts with the derivation of effective Hamiltonian for band electrons in magnetic field and continues with discussion of energy levels in magnetic field for massless Dirac fermions. The case of bilayer graphene is also considered. Special attention is paid to a formation of topologically protected zero-energy modes. Using band electrons in magnetic field as an example, a general concept of Berry phase is introduced. Magneto-oscillation effects for two-dimensional Dirac fermions are considered. Quantum Hall effect is discussed, via topological approach by Thouless and coworkers. At the end, we discuss electronic structure in the presence of crossed electric and magnetic fields and the effects of a smooth disorder on Landau levels for massless Dirac electrons.