Symmetric Skyrmions

Abstract

We present candidates for the global minimum energy solitons of charge one to nine in the Skyrme model, generated using sophisticated numerical algorithms. Assuming the Skyrme model accurately represents the low energy limit of QCD, these configurations correspond to the classical nuclear ground states of the light elements. The solitons found are particularly symmetric, for example, the charge seven skyrmion has icosahedral symmetry, and the shapes are shown to fit a remarkable sequence defined by a geometric energy minimization (GEM) rule. We also calculate the energies and sizes to within at least a few percent accuracy. These calculations provide the basis for a future investigation of the low energy vibrational modes of skyrmions and hence the possibility of testing the Skyrme model against experiment.

Solitons, links and knots

“…However, only in the last few years, with the advances in computing power, has it become possible to fully investigate topological solitons in three spatial dimensions. Recent results on Skyrmions [2,3,4] and monopoles (see for example, ref. [20]) have revealed that intricate and fascinating structures appear, which are inherently three-dimensional and therefore cannot arise as generalizations of solitons in one or two space dimensions.…”

Classical Skyrmions—static solutions and dynamics

“…Several were first discovered numerically [22], but they can now be rather well understood using the rational map ansatz for Skyrmions, reviewed in Section 4. For the energies of the Skyrmions see [9,11,12]. These energies satisfy the Faddeev-Bogomolny energy bound, E ≥ 12π 2 |B| [26], but equality is not attained for any field configuration with non-zero B [41].…”

Recent mathematical developments in the Skyrme model

“…3 Although it is not certain that this state is the absolute lowest energy state of the B = 2 sector, it does seem reasonable that it is so. No other states with lower energy have been found so far, numerical simulations of Skyrmion-Skyrmion scattering have not uncovered another intermediate state [92], and numerous modes of oscillation around the toroid have been considered without nding any negative modes [62]. We will assume here that it is in fact the state with lowest energy and that the manifold of low energy dynamics of the B = 2 sector consists of path joining the minimum and the two di erent families of saddle points.…”

“…On the other hand, this state is expected to take an active part in the scattering processes of two Skyrmions, especially those with zero impact parameter and with the relative orientation which generates maximum attraction. In such a situation, the Skyrmions move toward each other, deform and come together in the toroidal state before dividing into two Skyrmions which depart along a direction perpendicular to the initial one (the scattering plane is xed by the initial (iso)orientations of the Skyrmions) [53,92] (see gure 27 taken from reference [92]). This 90 scattering is typical of head on collisions of identical solitons [93].…”

“…Evidently, i f is invertible then 1 : ! S (92) de nes a push-forward and a pull-back in the opposite sense. We n o w come to the point of computing images of the metric and the inverse metric under and under its inverse (if it exists).…”