Discrete Symmetry Enhancement in Nonabelian Models and the Existence of Asymptotic Freedom

TL;DR

This paper demonstrates that a discrete icosahedral spin model and the continuous O(3) model share the same continuum limit, challenging the traditional view that the O(3) model is asymptotically free.

Contribution

It provides numerical evidence of universality between discrete and continuous models, showing the O(3) model's non-asymptotic freedom contrary to perturbative predictions.

Findings

Both models have the same continuum limit.

The icosahedral model is not asymptotically free.

Challenges perturbation theory predictions.

Abstract

We study the universality between a discrete spin model with icosahedral symmetry and the O(3) model in two dimensions. For this purpose we study numerically the renormalized two-point functions of the spin field and the four point coupling constant. We find that those quantities seem to have the same continuum limits in the two models. This has far reaching consequences, because the icosahedron model is not asymptotically free in the sense that the coupling constant proposed by L"uscher, Weisz and Wolff [1] does not approach zero in the short distance limit. By universality this then also applies to the O(3) model, contrary to the predictions of perturbation theory.