Unexpected results in asymptotically free quantum field theories

TL;DR

This paper investigates how replacing continuous symmetry groups with discrete subgroups in asymptotically free models affects their universality class and cut-off effects, revealing that large subgroups preserve the original models' universality.

Contribution

It provides precise numerical evidence that models with large discrete subgroups remain in the same universality class as the original AF models, and analyzes the nature of cut-off effects.

Findings

Models with large subgroups are in the same universality class as original models.

Cut-off effects follow an effective O(a) behavior up to correlation lengths of ~300.

Small statistical errors enable detailed analysis of these effects.

Abstract

We study the behavior of asymptotically free (AF) spin and gauge models when their continuous symmetry group is replaced by different discrete non-Abelian subgroups. Precise numerical results with relative errors down to O(0.1%) suggest that the models with large subgroups are in the universality class of the underlying original models. We argue that such a scenario is consistent with the known properties of AF theories. The small statistical errors allow a detailed investigation of the cut-off effects also. At least up to correlation lengths ~300 they follow effectively an O(a) rather than the expected O(a^2) form both in the O(3) and in the dodecahedron model.