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

Abstract

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 φ 2 1 , 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 C 1 , C −1 corresponding to the preimages of the points φ = (±1, 0, 0), respectively. Then the Hopf invariant can be defined as Q = link(C 1 , 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 C 1 and C −1 .

Fractional Skyrmion molecules in a $\mathbb{C}P^{N-1}$ model

“…Since Hopfions are topologically unstable in CP N −1 models due to π 3 (CP N −1 ) = 0 except for N = 2, this is a highly nontrivial dynamical question. If they are stable in a certain parameter region, they would be fractional Hopfion molecules, which generalize the fractional Hopfions in the usual Faddeev-Skyrme model with a symmetrybreaking potential [39], possibly of the form of closed fractional lump-strings linking each other.…”

Fractional Skyrmion molecules in a ℂPN−1 model

“…Since Hopfions are topologically unstable in CP N −1 models due to π 3 (CP N −1 ) = 0 except for N = 2, this is a highly nontrivial dynamical question. If they are stable in a certain parameter region, they would be fractional Hopfion molecules, which generalize the fractional Hopfions in the usual Faddeev-Skyrme model with a symmetry-breaking potential [127], possibly of the form of closed fractional lump-strings linking each other.…”

A steady Euler flow on the 3-sphere and its associated Faddeev-Skyrme solution

“…Although the original model is defined for M = R 3 and with α 0 = 0 [10], the interest was spread to the case of compact manifolds M [1,4,15,28,31] and to the case with potential (α 0 = 0) [1,11,14,24]. In the case of M = R 3 (where we impose lim |x|→∞ ϕ(x) = (0, 0, 1)) and of M = S 3 , since π 3 (S 2 ) ∼ = Z, the homotopy classes of maps ϕ : M → S 2 are indexed by the Hopf invariant Q(ϕ) = 1 ( S 2 ω) 2 S 3 α ∧ ϕ * ω, where ω is an area form on the codomain and α is any 1-form satisfying dα = ϕ * ω.…”