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

A Samoilenka,Ya Shnir

doi:10.1007/jhep09(2017)029

Abstract

We construct new solutions of the Faddeev-Skyrme model with a symmetry breaking potential admitting S1 vacuum. It includes, as a limiting case, the usual SO(3) symmetry breaking mass term, another limit corresponds to the potential m2ϕ12, 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 overrightarrow{phi}=left(pm 1,0,0right) , respectively. Then the Hopf invariant can be defined as Q=mathrm{link}left({mathcal{C}}_1,{mathcal{C}}_{-1}right) . 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} .

Highlights

The character of interaction between them, strongly depends to the form of the potential
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
In this paper we investigate the effect of a symmetry breaking potential admitting S1 vacuum on the soliton solutions of the Faddeev-Skyrme model

Summary

The model

For the relatively large values of the mass parameter m the location of the soliton can be identified as collection of curves of maximal energy, it interpolates between the preimages of antipodal point φ = (0, 0, −1) as above for α = 0 to the loops, which follow the preimages of two distinct points φ = (±1, 0, 0), as α = π/2: C1 = φ−1(1, 0, 0) and C−1 = φ−1(−1, 0, 0) Since these loops are linked Q times, the definition of the linking number can be related with the positions of the preimages of these points: Q = link(C1, C−1). The internal rotations of the triplet φ allow us to consider the limiting symmetry breaking potentials V ∼ φ12, ∼ φ22 and ∼ φ32 all on equal footing

Initial approximation

Numerical results

Degrees 3 and 4

Compact solutions

Findings

Summary