Knots and planar Skyrmions

Abstract

In this thesis the research presented relates to topological solitons in (2+1) and (3+1)-dimensional Skyrme theories. Solutions in these theories have topologically invariant quantities which results in stable solutions which are topologically distinct from a vacuum.

In Chapter 2 we discuss the broken baby Skyrme model, a theory which breaks symmetry to the dihedral group D_N. It has been shown that the unit soliton solution of the theory is formed of N distinct peaks, called partons. The multi-soliton solutions have already been numerically simulated for N = 3 and were found to be related to polyiamonds. We extend this for higher values of N and demonstrate that a polyform structure continues. We discuss our numerical simulations studying the dynamics of this model and show that the time dependent behaviour of solutions in the model can be understood by considering the interactions of individual pairs of partons. Results of these dynamics are then compared with those of the standard baby Skyrme model.

Recently it has been demonstrated that Skyrmions of a fixed size are able to exist in theories without a Skyrme term so long as the Skyrmion is located on a domain wall. In Chapter 3 we present a (2+1)-dimensional O(3) sigma model, with a potential term of a particular form, in which such Skyrmions exist. We numerically compute domain wall Skyrmions of this type. We also investigate Skyrmion dynamics so that we can study Skyrmion stability and the scattering of multi-Skyrmions. We consider scattering events in which Skyrmions remain on the same domain wall and find they are effectively one-dimensional. At low speeds these scatterings are well-approximated by kinks in the integrable sine-Gordon model. We also present more exotic fully two-dimensional scatterings in which Skyrmions initially on different domain walls emerge on the same domain wall.

The Skyrme-Faddeev model is a (3+1)-dimensional non-linear field theory that has topological soliton solutions, called hopfions. Solutions of this theory are unusual in that that they are string-like and take the form of knots and links. Solutions found to date take the form of torus knots and links of these. In Chapter 4 we show results which address the question of whether any non-torus knot hopfions exist. We present a construction of fields which are knotted in the form of cable knots to which an energy minimisation scheme can be applied. We find static hopfions of the theory which do not have the form of torus knots, but instead take the form of cable and hyperbolic knots.

In Chapter 5 we consider an approximation to the Skyrme-Faddeev model in which the soliton is modelled by elastic rods. We use this as a mechanism to study examples of particular knots to attempt to gain an understanding of why such knots have not been found in the Skyrme-Faddeev model. The aim of this study is to focus the search for appropriate rational maps which can then be applied in the Skyrme-Faddeev model.

The material presented in this thesis relates to two published papers and corresponding to Chapters 2 and 3 respectively, which were done as part of a collaboration. In this thesis my own results are presented. Chapter 4 concerns material which relates to the preprint which is all my own work. Chapter 5 discusses my own ongoing work.