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

TL;DR

This paper compares theoretical and simulation-based beta functions in two-dimensional non-linear sigma models, revealing the significant impact of topological defects on phase transitions and challenging assumptions in the 2+epsilon renormalization group approach.

Contribution

It demonstrates that topological defects influence phase transition behavior, showing discrepancies between theory and simulations in models with non-trivial topology.

Findings

Theoretical and measured beta functions agree for models with trivial topology.

Disagreement occurs in models with topological defects, indicating their importance.

Models with topological defects suggest phase transitions at finite temperature.

Abstract

For various two dimensional non linear $\sigma$ models, we present a direct comparison between the $\beta$ functions computed with the $2+\epsilon$ renormalization group and the $\beta$ functions measured by Monte Carlo simulations. The theoretical and measured $\beta$ 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 $2+\epsilon$ RG computation.