O(N) and RP^{N-1} Models in Two Dimensions

TL;DR

This paper demonstrates the equivalence of 2D $RP^{N-1}$ and $O(N)$ models for $N \\ge 3$ in the continuum limit, establishing they belong to the same universality class through analytical and numerical evidence.

Contribution

It provides a proof of equivalence between $RP^{N-1}$ and $O(N)$ models at weak coupling and clarifies their universality class in two dimensions.

Findings

Constraint models of $RP^{N-1}$ and $O(N)$ are equivalent at weak coupling.

Numerical results support that both models share the same universality class.

Finite size scaling differences are explained as boundary effects.

Abstract

I provide evidence that the 2D $RP^{N-1}$ model for $N \ge 3$ is equivalent to the $O(N)$-invariant non-linear $\sigma$-model in the continuum limit. To this end, I mainly study particular versions of the models, to be called constraint models. I prove that the constraint $RP^{N-1}$ and $O(N)$ models are equivalent for sufficiently weak coupling. Numerical results for their step-scaling function of the running coupling $\bar{g}^2= m(L) L$ are presented. The data confirm that the constraint $O(N)$ model is in the samei universality class as the $O(N)$ model with standard action. I show that the differences in the finite size scaling curves of $RP^{N-1}$i and $O(N)$ models observed by Caracciolo et al. can be explained as a boundary effect. It is concluded, in contrast to Caracciolo et al., that $RP^{N-1}$ and $O(N)$ models share a unique universality class.