Importance of the topological defects for two dimensional phase transitions and their relevance for the renormalization group

Abstract

For various two dimensional non linear $σ$ models, we present a direct comparison between the $β$ functions computed with the $2+ε$ renormalization group and the $β$ functions measured by Monte Carlo simulations. The theoretical and measured $β$ functions match each other nicely for models with a trivial topology, yet they disagree clearly for models containing topological defects. In these later cases, they are compatible with a phase transition at a finite temperature. This indicates that the global properties of the manifold do matter, in contradiction with the assumption used in the $2+ε$ RG computation.