Lesson 1: Basic experiments and phenomenology of superconductors in general. Meissner effect, the energy gap, demonstration of the lattice interaction (phonons). BCS theory, granular superconductors, Josephson effect. Why is it important to increase Tc? Technological and potential applications. Lesson 2: Introduction to high-temperature cuprate superconductors (HTSCs). Unconventional properties of HTSCs such as the pseudogap and charge density waves (CDWs). Experimental techniques; ARPES, STM, in addition to the usual ones such as susceptibility and resistivity measurements in HTSCs. Theory based on the superconducting phase fluctuation of Emery and Kivelson. The proposal of phase separation as an explanation for intrinsic disorder and some results; superfluid density, proximity effect, Hall effect, and the phase diagram.

The course aims to introduce and deepen the understanding of experimental methods employed in the investigation of superconducting materials, with emphasis on synthesis techniques, structural characterization, and measurements of electrical and magnetic properties. Practical aspects of sample preparation (ceramics, thin films, and single crystals), heat treatment, and purity control will be addressed, along with an introduction to cryogenic instrumentation and the use of high-sensitivity measurement devices. Theoretical lectures will be complemented by laboratory activities focused on determining the critical temperature (Tc), performing electrical resistance measurements under different magnetic fields and temperatures, recording magnetization curves (ZFC/FC), and estimating critical parameters (Hc, Jc). The course will also discuss experimental challenges related to strongly correlated systems and unconventional superconductors, highlighting the importance of careful data analysis and experimental reproducibility. The course is intended for graduate students in Physics and related fields who wish to develop practical skills in the characterization of superconducting materials, offering an integrated training that bridges theoretical foundations and advanced experimental procedures.

The optical conductivity is the extension to optical frequencies of the electrical resistivity of materials. It carries a quantitative
information on free and bound charge carriers. The total optical
spectral weight is a constant for a given material, which allows for the
analysis of coupling between excitations. In this short course I will show the basis of the optical conductivity and discuss the main models explaining optically active excitations (phonons, mobile carriers, interband transitions, gaps, etc). I will give a particular emphasis to the behavior of free carriers, from the basic Drude model to extended versions of it that take into account the quasi-particle self-energy and frequency dependent life-time effects. I will discuss the optical response observed in high temperature superconductors (cuprates and
pnictides), both in the strongly correlated normal state and the exotic superconducting phase. I’ll discuss features such as the pseudogap, the pairing mechanism, non-Fermi liquid signatures, and mass renormalization.

Here we point to some common features between the topological insulators and the type II superconductors that suggest their unification under a single theoretical framework. These common features become evident once the description of the topological insulators through the so-called first order equations is taken. These equations were firstly found by Alexei Abrikosov in his 1957 seminal work where the magnetic properties of superconducting alloys were understood. A deeper understanding of the first order equations was later obtained by Evgeniy Bogomolny when he proved that they solved exactly the Ginzburg-Landau theory for one value of the coupling constant. Here we propose that such equations describe the topological insulators once spin is included into them. Undoubtedly the Abrikosov-Bogolmony (AB) equations provide a venue to understand the topological properties of a system through its magnetic properties. The simplest set of such equations is associated to the complex numbers and describes the type II superconductors. They give rise to vortices and they are two dimensional, that is, they are applicable to three-dimensional space with uniaxial symmetry. The inclusion of spin (quaternions), renders the equations three-dimensional. Nevertheless, a closer investigation shows that they describe surface quantum plasmons, and so, apply to conducting layers where topological skyrmions arise to play the role of the fundamental quasi-particles. The AB set consists of two equations, and in case of spin, the first one is just a zero-helicity condition that guarantees the spin-momentum locking, a feature commonly found in the topological insulators. The second spinorial AB equation determines the local magnetic field which is responsible for the topological stability of the skyrmionic quasi-particles through the Chern-Simons index. The AB equations accept a Lorentz generalization to four dimensions, known as the Seiberg-Witten equations that describe massless monopoles. The AB equations are intimately related to the so-called Lichnerowicz-Weitzenbock decomposition of the kinetic energy, that here is extended to three terms to include a surface effect, which is shown to be exactly the Rashba interaction. As an outcome, the spectrum of the zero-helicity state is found to be linear. Hence the present framework provides a natural explanation for the onset of the Rashba term and of its Dirac like spectrum, which are common features of the topological insulators. For the case of several stacked layers, the arousal of a pseudo gap, the presence of conducting bands with different masses, the topological stability of quasi-particles in the form of skyrmions and anti-skyrmions are straighforward consequences of the present approach.