Static solitons with non-zero Hopf number

TL;DR

This paper studies a generalized non-linear sigma model in three dimensions, focusing on static solitons characterized by the Hopf number, and explores their properties, energies, and effects of additional potentials.

Contribution

It provides explicit constructions of static solitons with Hopf charges 1 and 2, analyzes their shapes and energies, and examines the impact of potential terms and slow rotations.

Findings

Explicit soliton solutions for Hopf charge 1 and 2

Bound on soliton energies based on topology

Approximate spectrum for rotating solitons

Abstract

We investigate a generalized non-linear O(3) $\sigma$-model in three space dimensions where the fields are maps $S^3 \mapsto S^2$. Such maps are classified by a homotopy invariant called the Hopf number which takes integer values. The model exhibits soliton solutions of closed vortex type which have a lower topological bound on their energies. We explicitly compute the fields for topological charge 1 and 2 and discuss their shapes and binding energies. The effect of an additional potential term is considered and an approximation is given for the spectrum of slowly rotating solitons.