Hopf Solitons in the AFZ Model

TL;DR

This paper numerically investigates topological solitons in the AFZ model, revealing various static solutions including knots and links, and explores their behavior across related conformal Skyrme-Faddeev models.

Contribution

It provides the first comprehensive numerical analysis of axial, knot, and linked solitons in the AFZ model and examines their transitions in related models.

Findings

Static solutions include a trefoil knot at Hopf index five.

Transition from linked to axial solutions occurs at Hopf index four.

Fewer models support axial solitons compared to linked at certain indices.

Abstract

The Aratyn-Ferreira-Zimerman (AFZ) model is a conformal field theory in three-dimensional space. It has solutions that are topological solitons classified by an integer-valued Hopf index. There exist infinitely many axial solutions which have been found analytically. Axial, knot and linked solitons are found numerically to be static solutions using a modified volume preserving flow for Hopf index one to eight, allowing for comparison with other Hopf soliton models. Solutions include a static trefoil knot at Hopf index five. A one-parameter family of conformal Skyrme-Faddeev (CSF) models, consisting of linear combinations of the Nicole and AFZ models, are also investigated numerically. The transition of solutions for Hopf index four is mapped across these models. A topological change between linked and axial solutions occurs, with fewer models permitting axial solitons than linked solitons at Hopf index four.