Isospinning hopfions

TL;DR

This paper numerically investigates isospinning hopfions in the Faddeev-Skyrme model, revealing that their shape remains mostly unchanged with rotation frequency while their size increases, and introduces an elastic rod model to explain these behaviors.

Contribution

It reformulates the construction of rotating solitons as a variational problem and develops a simple elastic rod model to explain numerical findings.

Findings

Soliton shape is mostly independent of rotation frequency.

Soliton size increases monotonically with frequency.

Elastic rod model effectively captures numerical results.

Abstract

The problem of constructing internally rotating solitons of fixed angular frequency $\omega$ in the Faddeev-Skyrme model is reformulated as a variational problem for an energy-like functional, called pseudoenergy, which depends parametrically on $\omega$. This problem is solved numerically using a gradient descent method, without imposing any spatial symmetries on the solitons, and the dependence of the solitons' energy on $\omega$, and on their conserved total isospin $J$, studied. It is found that, generically, the shape of a soliton is independent of $\omega$, and that its size grows monotonically with $\omega$. A simple elastic rod model of time-dependent hopfions is developed which, despite having only one free parameter, accounts well for most of the numerical results.