Geometrical and topological study of the Kosterlitz-Thouless phase transition in the XY model in two dimensions

Phase transitions do not necessarily correspond to a symmetry-breaking phenomenon. This is the case of the Kosterlitz-Thouless (KT) phase transition in a two-dimensional classical XY model, a typical example of a transition stemming from a deeper phenomenon than a symmetry-breaking. Actually, the KT transition is a paradigmatic example of the successful application of topological concepts to the study of phase transition phenomena in the absence of an order parameter. Topology conceptually enters through the meaning of defects in real space. In the present work, the same kind of KT phase transition in a two-dimensional classical XY model is tackled by resorting again to a topological viewpoint, however focussed on the energy level sets in phase space rather than on topological defects in real space. Also from this point of view, the origin of the KT transition can be attributed to a topological phenomenon. In fact, the transition is detected through peculiar geometrical changes of the energy level sets which, after a theorem in differential topology, are direct probes of topological changes of these level sets.