Hopf solitons and elastic rods

TL;DR

This paper introduces an elastic rod model to approximate Hopf solitons in the Skyrme-Faddeev model, capturing key features like buckling, linking, and knot formation with a simplified energy formulation.

Contribution

It develops an elastic rod approximation derived from the field theory energy, extending the classical Kirchhoff model to describe topological solitons.

Findings

The model reproduces buckling of charge three solutions.

It captures linking phenomena at charges five and six.

The minimal energy trefoil knot at charge seven is also modeled.

Abstract

Hopf solitons in the Skyrme-Faddeev model are string-like topological solitons classified by the integer-valued Hopf charge. In this paper we introduce an approximate description of Hopf solitons in terms of elastic rods. The general form of the elastic rod energy is derived from the field theory energy and is found to be an extension of the classical Kirchhoff rod energy. Using a minimal extension of the Kirchhoff energy, it is shown that a simple elastic rod model can reproduce many of the qualitative features of Hopf solitons in the Skyrme-Faddeev model. Features that are captured by the model include the buckling of the charge three solution, the formation of links at charges five and six, and the minimal energy trefoil knot at charge seven.