The Skyrme Model: Curved
Space, Symmetries and Mass

Abstract

The presented thesis contains research on topological solitons in (2+1) and (3+1) dimensional classical field theories, focusing upon the Skyrme model. Due to the highly non-linear nature of this model, we must consider various numerical methods to find solutions.

We initially consider the (2+1) baby Skyrme model, demonstrating that the currently accepted form of minimal energy solutions, namely straight chains of alternating phase solitons, does not hold for higher charge. Ring solutions with relative phases changing by pi for even configurations or pi-pi/B for odd numbered configurations, are demonstrated to have lower energy than the traditional chain configurations above a certain charge threshold, which is dependant on the parameters of the model. Crystal chunk solutions are then demonstrated to take a lower energy but for extremely high values of charge. We also demonstrate the infinite charge limit of each of the above configurations. Finally, a further possibility of finding lower energy solutions is discussed in the form of soliton networks involving rings/chains and junctions. The dynamics of some of these higher charge solutions are also considered.

In chapter 3 we numerically simulate the formation of (2+1)-dimensional baby Skyrmions from domain wall collisions. It is demonstrated that Skyrmion, anti-Skyrmion pairs can be produced from the interaction of two domain walls, however the process can require quite precise conditions. An alternative, more stable, formation process is proposed and simulated as the interaction of more than two segments of domain wall. Finally domain wall networks are considered, demonstrating how Skyrmions may be produced in a complex dynamical system.

The broken planar Skyrme model, presented in chapter 4, is a theory that breaks global O(3) symmetry to the dihedral group D_N. This gives a single soliton solution formed of N constituent parts, named partons, that are topologically confined. We show that the configuration of the local energy solutions take the form of polyform structures (planar figures formed by regular N-gons joined along their edges, of which polyiamonds are the N=3 subset). Furthermore, we numerically simulate the dynamics of this model.

We then consider the (3+1) SU(2) Skyrme model, introducing the familiar concepts of the model in chapter 5 and then numerically simulating their formation from domain walls. In analogue with the planar case, it is demonstrated that the process can require quite precise conditions and an alternative, more stable, formation process can be achieved with more domain walls, requiring far less constraints on the initial conditions used.

The results in chapter 7 discuss the extension of the broken baby Skyrme model to the 3-dimensional SU(2) case. We first consider the affect of breaking the isospin symmetry by altering the tree level mass of one of the pion fields breaking the SO(3) isospin symmetry to an SO(2) symmetry. This serves to exemplify the constituent make up of the Skyrme model from ring like solutions. These rings then link together to form higher charge solutions. Finally the mass term is altered to allow all the fields to have an equivalent tree level mass, but the symmetry of the Lagrangian to be broken, firstly to a dihedral symmetry D_N and then to some polyhedral symmetries.

We now move on to discussing both the baby and full SU(2) Skyrme models in curved spaces. In chapter 8 we investigate SU(2) Skyrmions in hyperbolic space. We first demonstrate the link between increasing curvature and the accuracy of the rational map approximation to the minimal energy static solutions. We investigate the link between Skyrmions with massive pions in Euclidean space and the massless case in hyperbolic space, by relating curvature to the pion mass. Crystal chunks are found to be the minimal energy solution for increased curvature as well as increased mass of the model. The dynamics of the hyperbolic model are also simulated, with the similarities and differences to the Euclidean model noted.

One of the difficulties of studying the full Skyrme model in (3+1) dimensions is a possible crystal lattice. We hence reduce the dimension of the model and first consider crystal lattices in (2+1)-dimensions. In chapter 9 we first show that the minimal energy solutions take the same form as those from the flat space model. We then present a method of tessellating the Poincare disc model of hyperbolic space with a fundamental cell. The affect this may have on a resulting Skyrme crystal is then discussed and likely problems in simulating this process.

We then consider the affects of a pure AdS background on the Skyrme model, starting with the massless baby Skyrme model in chapter 10. The asymptotics and scale of charge 1 massless radial solutions are demonstrated to take a similar form to those of the massive flat space model, with the AdS curvature playing a similar role to the flat space pion mass. Higher charge solutions are then demonstrated to exhibit a concentric ring-like structure, along with transitions (dubbed popcorn transitions in analogy with models of holographic QCD) between different numbers of layers. The 1st popcorn transitions from an n layer to an n+1-layer configuration are observed at topological charges 9 and 27 and further popcorn transitions for higher charges are predicted. Finally, a point-particle approximation for the model is derived and used to successfully predict the ring structures and popcorn transitions for higher charge solitons.

The final chapter considers extending the results from the penultimate chapter to the full SU(2) model in a pure AdS_4 background. We make the prediction that the multi-layered concentric ring solutions for the 2-dimensional case would correlate a multi-layered concentric rational map configuration for the 3-dimensional model. The rational map approximation is extended to consider multi-layered maps and the energies demonstrated to reduce the minimal energy solution for charge B=11 which is again dubbed a popcorn transition. Finally we demonstrate that the multi shell structure extends to the full field solutions which are found numerically. We also discuss the affect of combined symmetries on the results which (while likely to be important) appear to be secondary to the dominant effective potential of the metric which simulates a packing problem and hence forces the popcorn transitions to act accordingly with the 2-dimensional model.