Universality Class of $O(N)$ Models

TL;DR

The paper discusses the critical behavior of the two-dimensional $O(3)$ model, challenging the expected asymptotic freedom and suggesting a different critical exponent based on numerical data.

Contribution

It highlights inconsistencies in the critical exponent of the $O(3)$ model, questioning the conventional understanding of its universality class.

Findings

Numerical data suggest $oxed{ ext{η}=1/4}$ for the $O(3)$ model.

This value conflicts with the asymptotic freedom hypothesis.

The critical exponent differs from that of the Wess-Zumino-Novikov-Witten model at $ heta= ext{π}$.

Abstract

We point out that existing numerical data on the correlation length and magnetic susceptibility suggest that the two dimensional $O(3)$ model with standard action has critical exponent $\eta=1/4$, which is inconsistent with asymptotic freedom. This value of $\eta$ is also different from the one of the Wess-Zumino-Novikov-Witten model that is supposed to correspond to the $O(3)$ model at $\theta=\pi$.