Asymptotically free models and discrete non-Abelian groups

TL;DR

This paper investigates how perturbations reducing O(3) symmetry to discrete Platonic symmetries affect two-dimensional sigma models, estimating correlation lengths and explaining recent numerical findings.

Contribution

It introduces a detailed analysis of symmetry-breaking perturbations in sigma models and provides estimates for correlation lengths where model behaviors diverge.

Findings

Correlation length for icosahedron model exceeds 200

Explains recent numerical results by Patrascioiu and Seiler, Hasenfratz and Niedermayer

Shows asymptotic freedom persists under certain discrete symmetry perturbations

Abstract

We study the two-dimensional renormalization-group flow induced by perturbations that reduce the global symmetry of the O(3) sigma-model to the discrete symmetries of Platonic solids. We estimate the value of the correlation length at which differences in the behaviour of the various models should be expected. For the icosahedron model, we find xi > 200. We provide an explanation for the recent numerical results of Patrascioiu and Seiler and of Hasenfratz and Niedermayer.