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

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 $m^2{\varphi }_1^2$, 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 $\vec{\varphi }=(\pm 1\mp \mu ,0,0)$, respectively. Then the Hopf invariant can be defined as $Q=\mathrm{link}({\mathcal{C}}_1,{\mathcal{C}}_{-1})$. 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 ${\mathcal{C}}_{-1}$.

22 pages, 129 figures

References in corpus (7)

• Skyrmion knots in frustrated magnets
• Skyrmions, instantons and holography
• Twisted toroidal vortex-solitons in inhomogeneous media with repulsive nonlinearity
• Knots in the Skyrme-Faddeev model
• Non-Meissner electrodynamics and knotted solitons in two-component superconductors
• Rotational symmetry breaking in baby Skyrme models
• Gauged Baby Skyrme Model with Chern-Simons term

Cited by in corpus (2)

• Fractional Skyrmion molecules in a $\mathbb{C}{P}^{N-1}$ model
• Fractional solitons in non-Euclidian elastic plates