Asymptotically free theories based on discrete subgroups

TL;DR

This paper investigates the critical behavior of discrete spin models related to the 2d O(3) non-linear sigma model, showing that models with large discrete subgroups share the same universality class as the original model, with cut-off effects diminishing linearly with lattice spacing.

Contribution

It provides numerical evidence that large discrete subgroups produce models in the same universality class as the continuous O(3) model, extending understanding of discretization effects.

Findings

Models with large discrete subgroups match the universality class of the O(3) model.

Cut-off effects decrease approximately linearly with lattice spacing up to correlation length 300.

Discrete models exhibit similar critical behavior to the continuous sigma model.

Abstract

We study the critical behavior of discrete spin models related to the 2d O(3) non-linear sigma model. Precise numerical results suggest that models with sufficiently large discrete subgroups are in the same universality class as the original sigma model. We observe that at least up to correlation lengths $\xi\approx 300$ the cut-off effects follow effectively an $\propto a$ behaviour both in the O(3) and in the dodecahedron model.