Band theory describes electrons as independent quantum particles which interact with the ionic lattice. Interactions are responsible for the insulating behavior of some materials which are expected to be metallic or for the increase of the resistivity with decreasing temperature in metals with magnetic impurities. In many cases interactions tend to localize the electrons at the lattice sites. They are also behind the appearance of magnetic or superconducting phases. Recently it has become clear that besides interactions topology can play an important role on the electronic properties and materials with non trivial topological properties have been discovered.

In the course we will justify the success of band theory (Fermi liquid theory) and describe the formation of local moments due to interactions (Mott physics and Hund metals). The interplay between itinerant quasiparticles and local moments will be addressed when addressing the Kondo effect, heavy fermions, and the magnetic tendencies. We will discuss how the state of the system changes at a classical or quantum phase transition and ordered phases such as the superconducting state.  We will give an introduction to Dirac materials and to the topological phases of matter: topological insulators, Weyl semimetals and topological superconductivity,  as well as to non-equilibrium physics.

To follow properly the course it is necessary to have good knowledge of quantum mechanics, solid state physics, Fermi-Dirac statistics and second quantization.