Solitons, links and knots

Abstract

Using numerical simulations of the full nonlinear equations of motion we investigate topological solitons of a modified O(3) sigma model in three space dimensions, in which the solitons are stabilized by the Hopf charge. We find that for solitons up to charge five the solutions have the structure of closed strings, which become increasingly twisted as the charge increases. However, for higher charge the solutions are more exotic and comprise linked loops and knots. We discuss the structure and formation of these solitons and demonstrate that the key property responsible for producing such a rich variety of solitons is that of string reconnection.

Hopf solitons on the lattice

“…In these pictures, the lattice spacing is unity; so one sees that the solitons are typically spread over a width of 10-20 lattice units (depending on their shape and on the value of H). It is remarkable that the pictures closely resemble the corresponding ones in the continuum case [8,10], despite the systems being quite different.…”

The interaction of two Hopf solitons

“…The two-ring structure is evident from (10). Minimizing the energy of (10) with respect to the ten parameters (α, β, and the coefficients of f and h) gives an energy E = 2.29, it ie 1.3% above that of the solution.…”

“…As depicted in Fig 2, the two rings (where W = ∞) merge into one, and then the two lines where W = 0 merge as well. We end up with a solution which has been described previously [7], [9], [10], and which is believed to be the global minimum in the N = 2 sector. It is axially-symmetric, and resembles a single ring; but this time the ring winds around a double copy of the z-axis, and hence it has a linking number of N = 2.…”

“…This leads to a sigma model which admits stable, static, localized solitons -they resemble closed strings, which can be linked or knotted. The system has been written about at least since 1975 [4], [5]; but recently interest in it has increased, stimulated by numerical work [6], [7], [8], [9], [10], [11]; see also [12], [13], [14].…”

Particle creation via relaxing hypermagnetic knots

“…Geometrically, B is tangent to the closed curves χ = const (see e.g. [23,25,26,24]). The Hopf index N H of χ may be computed from B via…”