Magnetic-superconducting phase coexistence driven by on-site repulsions in BCS-like models

Abstract

This thesis focuses on the study of the thermodynamics of fermionic systems on lattices, from the point of view of the C*-algebraic formulation of quantum mechanics, and on the application of the so-called "catastrophe theory" to the analysis of the phase diagram of an explicit fermionic model. The first chapter is devoted to the introduction of some of the basic and most important properties of C*-algebras, and also to the study of its representations and its set of states. The second chapter is devoted to the development of some important results of catastrophe theory. The results derived in the chapter allows one to analyze the behavior of the minima of members of a family of functions around a degenerate critical point, and they will be used to study the behavior of the thermodynamic pressure for a given fermionic lattice model. The third chapter presents some of the formalism developed in [4], that concerns the existence of the thermodynamics and equilibrium states of lattice fermi systems subject to a suitable set of long-range interactions (i.e., interactions containing mean-field terms). Finally, in the last chapter, using all the formalisms and results already studied in the previous chapters, the thermodynamics of an explicit BCS-like model is analyzed. With the help of catastrophe theory, it is shown that such model exhibits a coexistence of magnetic and superconducting phases for a suitable choice of parameters, and through a perturbative analysis it is also shown that the coexistence still holds when a small kinetic term is added into the model.