Phase Separation and Interfaces. Exact Results

We will devote Chapter 1 to a short review of traditional approaches to interfacial phenomena. This starts with an overview on phenomenological descriptions and terminates with a discussion on mean field theories of interfaces. In Chapter 2 we recall some essential notions of scattering theory in two dimensions on which we will rely in the rest of the thesis. In Chapter 3 we will pose the basis of the exact field-theoretic approach to phase separation in two dimensions. In particular, we will develop the formalism for the study of interfaces in a strip geometry. Drops on a flat substrate and the corresponding wetting transition will be discussed in Chapter 4. In Chapter 5 we will analyze phase separation in presence of a wedge-shaped substrate and its field-theoretical implications. The exposition will cover phase separation both with and without the occurrence of intermediate phases. These two regimes will be discussed in detail for the strip, half-plane and wedge geometries. Our study is based on universal properties of the scaling limit and accounts exactly for the properties of the different universality classes. The field-theoretical approach to near-critical behavior does not exhaust its applications to interfacial phenomena. We will conclude in Chapter 6 with a further application in which we will consider the thermal Casimir e↵ect, i.e. the analogue of the quantum Casimir e↵ect for statistical systems near criticality. We will show how bulk and boundary e↵ects, jointly with the symmetry of boundary conditions, play a role in the determination of the long-distance decay of the Casimir force.