Phase Diagrams of Systems Exhibiting Modulated Structures

This thesis deals with theoretical models of modulated structures. We do not try to reproduce experimental observations with accuracy, but are interested in the qualitative behavior of systems exhibiting modulated structures. Chapter 2 shows how an exact phase diagram can be computed for an exactly solvable model of the Frenkel -Kontorova type involving N parabolas. Upsilon points are present in the phase diagram of the two parabolas model, and the three parabolas model contains lines of upsilon points ending in new types of critical points. In the third and fourth chapters, we study the ANNNI model in the mean field approximation with the help of the effective potential method. The efficiency of this numerical method is improved, and the phase diagram of the ANNNI model found in this manner agrees with previous results. We also use the notions of defects to study the boundaries of the main commensurate phases. The last chapter concerns an XY spin model with second neighbor interactions. With the help of the effective potential method, and also by using some assumption about the sequence of phases, we draw phase diagrams for various cases: with or without a six-fold anisotropy, and with or without an external field. The case of an external field without anisotropy is of special interest since there exists a commensurate phase of period three which is sliding. We show that this phase is stable for only one value of the ratio of the interactions strengths, and is destabilized by a small twist as the ratio changes.