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This chapter provides a review of recently-developed Dynamical Mean-Field Theory (DMFT) approaches to the general problem of strongly correlated electronic systems with disorder. The chapter first describes the standard DMFT approach, which is exact in the limit of large coordination, and explain why in its simplest form it cannot capture either Anderson localization or the glassy behavior of electrons. Various extensions of DMFT are then described, including statistical DMFT, typical medium theory, and extended DMFT, methods specifically designed to overcome the limitations of the original formulation. The chapter provides an overview of the results obtained using these approaches, including the formation of electronic Griffiths phases, the self-organized criticality of the Coulomb glass, and the two-fluid behavior near Mott-Anderson transitions. Finally, the chapter outlines research directions that may provide a route to bridge the gap between the DMFT-based theories and the complementary diffusion-mode approaches to the metal-insulator

This Overview provides a general introduction to metal-insulator transitions, with focus on specific mechanisms that can localize the electrons in absence of magnetic or charge ordering, and produce well defined quantum critical behavior. The Overview contrasts the physical picture of Mott, who emphasized the role of electron-electron interactions, and that of Anderson, who stressed the possibility of impurity-induced bound state formation, as alternative routes to arrest the electronic motion. It also describes more complicated situations when both phenomena play coexist, leading to meta-stability, slow relaxation, and glassy behavior of electrons. A critical overview of the available theoretical approaches is then presented, contrasting the weak-coupling perspective, which emphasizes diffusion-mode corrections, and the strong-coupling viewpoint, which stresses inhomogeneous phases and local correlation effects. The Overview gives specific examples of experimental systems, providing clues on what should be the most profitable path forward in unraveling the mystery of metal-insulator transitions.

This chapter yields an introduction to quantum Hall effects both for non-relativistic electrons in conventional two-dimensional electron gases (such as in semiconductor heterostructures) and relativistic electrons in graphene. After a brief historical overview follows a detailed discussion of the kinetic-energy quantisation of non-relativistic and relativistic electrons in a strong magnetic field (section 2). Section 3 is devoted to the transport characteristics of the integer quantum Hall effect, and the basic aspects of the fractional quantum Hall effect are described in section 4. In section 5, several multicomponent quantum Hall systems are briefly discussed, namely the quantum Hall ferromagnetism, bilayer systems and graphene that may be viewed as a four-component system.

This chapter provides an introduction to the field of strong correlation physics with ultracold atoms in optical lattices. After a basic introduction to the single-particle band structure and lattice configurations, the effect of strong interactions on the Hubbard model is discussed. Detection methods are introduced, which allow us to reveal in-trap density and (quasi)-momentum distributions, as well as correlations between particles on the lattice. The fundamental phases of the bosonic and fermionic Hubbard model are discussed. Superexchange spin-spin interactions that form the basis of quantum magnetism are introduced and the current status on observing such magnetic phenomena is highlighted. Finally, novel possibilities for detecting single-site and single-atom resolved quantum gases are outlined.

This chapter focuses on the cuprates, which are uniquely interesting superconducting compounds due to their high Tc, peculiar properties, and potential for applications. The history of the discovery of this very unusual class of superconductors is described, together with the properties and key theoretical concepts that can be used to understand their superconducting and normal behaviours. This chapter contains a description of some very key aspects of these materials: their very unusual phase diagram, where doping takes the compounds from antiferromagnetic insulators to high-temperature superconductors and finally to metallic conductivity; their very anomalous upper critical field Hc2; the symmetry of their order parameter; and the unusual isotope effect on Tc and penetration depth. There are two main approaches to the issue of the origin of high Tc in the cuprates: the phonon mechanism, with the strong impact of the polaronic effect, and a mechanism based on strong correlation effects.