Generalized Migdal-Kadanoff Renormalization Group for Competing Interactions. Applications to 2-DIMENSIONAL Systems.

In the early 1970's a major development occurred in Statistical Mechanics which involved the application and refinement of a new technique called Renormalization Group (RG). The RG is a method by which a system with many degrees of freedom is simplified by the reduction of these degrees of freedom in a systematic manner, which keeps only those that are essential to the "physics" of the problem. The simplified system can then be treated with a variety of techniques. Since the pioneering work of Wilson the RG approach has become an essential tool in a wide variety of problems. Many types of RG schemes have been used to get information about systems in the neighborhood of a critical point. Each method has its advantages and limitations. In designing an RG technique there is no general rule. Each transformation must be tailored to the system under study. There are two classes of transformations: Position Space Renormalization Group (PSRG) and Momentum Space Renormalization Group (MSRG). In MSRG we keep the degrees of freedom that correspond to the central part of the Brillouin zone and we integrate over the others. In PSRG we sum out small length scale degrees of freedom. This approach is more direct and has had great success in problems involving discrete spins e.g. Ising models. Recently there has been increasing interest in systems where competing interactions play an essential role. As examples we can mention layers of atoms adsorbed in graphite and magnetic systems with ferromagnetic and antiferromagnetic interactions. There has been a lot of theoretical and experimental work in this field and new kinds of phase transitions has been observed. These systems display exciting new features such as exotic orderings, multicritical points and large number of phases. As the problems grow in complexity we must improve the RG methods. This work represents a step in this direction. We have generalized the Migdal-Kadanoff Renormalization Group in order to allow us to treat systems with competing interactions. As result we have been able to apply this method to problems for which the standard technique cannot be applied.