Geometric study of classical phase transitions and quantum entanglement

The present work is divided into two parts. First, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian of a system under investigation. In particular, it turns out that peculiar behaviors of thermodynamic observables at a phase transition point are rooted in more fundamental changes in the geometry of the energy level sets in phase space. More specifically, we discuss how microcanonical and geometrical descriptions of phase transitions are shaped in the peculiar Kosterlitz–Thouless phase transition and in the special case of the ϕ4 model. In the second part, we will focus on the quantification of quantum entanglement, a topic of great current interest. We will derive entanglement and quantum correlation measures, from a geometrical procedure, which are valid for multipartite hybrid states. We also provide a physical and operational meaning of the proposed entanglement measures for pure states. Furthermore, we show that the proposed measures can either be analytically or numerically computed. Finally, we test the validity of the proposed measure through a variety of examples.