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This review article describes how the simple Ginzburg-Landau approach lies in the heart of the general theory of superconductors. I t introduces the reader to the handling of the theory and to the numerous possibilities of applications. A general free energy functional for a superconductor is given and we study the different cases where it can be reduced to a Ginzburg-Landau form or to a simple generalization of this form. We emphasize that applications are not restricted to thermodynamical ones as the Ginzburg-Landau approach can be used in the calculation of dissipative phenomena. The possibility of an extension to the time-dependent phenomena is discussed in detail to present the difficult problems which arise in that case. The gapless regime of type I1 superconductors is given as an example.

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Résumé. 2014

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Below the critical temperature Tc ≃ 600K, an iso-structural transition, named γ-α transition, can be induced in Cerium by applying pressure. This transition is first-order, and is accompanied by a sizable volume collapse. A conclusive theoretical explanation of this intriguing phenomenon has still not been achieved, and the physical pictures proposed so far are still under debate. In this work, we illustrate zero-temperature first-principle calculations which clearly demonstrate that the γ-α transition is induced by the interplay between the electron-electron Coulomb interaction and the spin-orbit coupling. We address the still unresolved problem on the existence of a second low-T critical point, i.e., whether the energetic effects alone are sufficient or not to induce the γ-α transition at zero temperature. The γ-α iso-structural transition in Cerium [1] was discovered in 1949 [2]. Since then, a lot of theoretical and experimental work has been devoted to its understanding. The great interest in this phenomenon arises from the fact that the transition is isostructural, i.e., the lattice structure of the system is equal in the two phases. Furthermore, the possibility that the underlying mechanism lies in the electronic structure only-i.e., without it being necessary to involve other effects-makes Cerium a potential theoretical testing ground for basic concepts of correlated electron systems. Two main theoretical pictures are still under debate to explain the volume collapse: the Kondo volume collapse (KVC) [3, 4] and the orbital-selective Mott transition within the Hubbard model (HM) [5]. According to the KVC the transition is induced by the rapid change of the coherence temperature across the transition boundaries, which affect dramatically the structure of the conduction spd electrons through Kondo effect. According to the HM, instead, it is the hopping between f orbitals that changes drastically across the transition between the α phase (with delocalized f electrons) and the γ phase (with localized f electrons), as for the Mott transition in the Hubbard model. Consistently with both the HM and the KVC pictures, the f-electrons are strongly correlated both in the α and in the γ phase. This fact is clearly indicated, e.g., by the photoemission spectra, which is known experimentally [6-8] and theoretically [9-11]. Despite this similarity , there is a key difference between these two models: while the KVC attributes a very important role to the interplay between the localized 4f orbitals and the itinerant spd conduction bands, the itinerant electrons are "spectators" in the HM picture. The development of LDA+DMFT (Local Density Approximation plus Dynamical Mean Field Theory) [12] results [13-15] has successfully reproduced many aspects of this transition, and different aspects of these studies can be understood in both physical pictures. There are still fundamental questions which have not been answered. (1) What is the role of the spin-orbit interaction (SOC) for the volume-collapse? (2) What is the fate of the ...

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