Fractional Hopfions in the Faddeev-Skyrme model with a symmetry breaking potential

TL;DR

This paper constructs new fractional Hopfion solutions in the Faddeev-Skyrme model with a symmetry-breaking potential, revealing complex linked tube structures and knotted configurations characterized by a generalized Hopf invariant.

Contribution

It introduces a novel class of solutions with symmetry-breaking potentials, including knotted and linked tube structures, expanding the understanding of topological solitons in the Faddeev-Skyrme model.

Findings

Discovery of solutions with linked tube structures and trefoil knots.

Definition of a generalized Hopf invariant based on preimage loops.

Identification of solutions with different types of preimage curves.

Abstract

We construct new solutions of the Faddeev-Skyrme model with a symmetry breaking potential admitting $S^1$ vacuum. It includes, as a limiting case, the usual $SO(3)$ symmetry breaking mass term, another limit corresponds to the potential $m^2 \phi_1^2$, which gives a mass to the corresponding component of the scalar field. However we find that the spacial distribution of the energy density of these solutions has more complicated structure, than in the case of the usual Hopfions, typically it represents two separate linked tubes with different thicknesses and positions. In order to classify these configurations we define a counterpart of the usual position curve, which represents a collection of loops $\mathcal{C}_1, \mathcal{C}_{-1}$ corresponding to the preimages of the points $\vec \phi = (\pm 1 \mp \mu, 0,0)$, respectively. Then the Hopf invariant can be defined as $Q= {\rm link} (\mathcal{C}_1,\mathcal{C}_{-1})$. In this model, in the sectors of degrees $Q=5,6,7$ we found solutions of new type, for which one or both of these tubes represent trefoil knots. Further, some of these solutions possess different types of curves $\mathcal{C}_1$ and $\mathcal{C}_{-1}$.