Mathematics

The discrete structure of the ground states of a spin system is often neglected by introducing a mesoscopic scale that captures the main features of the problem while simplifying its analysis. In general this procedure is not rigorous and not even well understood. In this seminar we will show that in the case of the ground states of the XY (N dimensional possibly anysotropic) spin type models it is indeed possible to justify, from a variational point of view, the coarse graining procedure which leads to a Ginzburg-Landau type energy on a mesoscopic scale. Topological type phase transitions will be highlighted by showing the presence of vortice-like ground states. In the 2-dimesional case we will show how it is possible to address the problem in the case of long range interactions. Finally we will discuss several possible developments and present some open problems.