Quantization ofTd- andOh-symmetric Skyrmions

Abstract

The geometrical construction of rational maps using a cubic grid has led to many new Skyrmion solutions, with baryon numbers up to 108. Energy spectra of some of the new Skyrmions are calculated here by semi-classical quantization. Quantization of the B = 20 T d -symmetric Skyrmion, which is one of the newly found Skyrmions, is considered, and this leads to the development of a new approach to solving Finkelstein-Rubinstein (F-R) constraints. Matrix equations are simplified by introducing a Cartesian version of angular momentum basis states, and the computations are easier. The quantum states of all T d -symmetric Skyrmions, constructed from the cubic grid, are classified into three classes, depending on the contribution of vertex points of the cubic grid to the rational maps. The analysis is extended to the larger symmetry group O h . Quantum states of O h -symmetric Skyrmions, constructed from the cubic grid, form a subset of the T d -symmetric quantum states.

Classically isospinning Skyrmion solutions

“…Within our numerical accuracy the inertia tensors (8) computed with our 3D-relaxation code are all found to be proportional to the unit matrix with Λ ¼ 47.5. This is in reasonable agreement with the value calculated within the hedgehog ansatz (22) and with the moments of inertia [9] using a three-dimensional nonlinear conjugate gradient method.…”

“…Note that the moments of inertia U 33 ¼ 68.9; V 33 ¼ 275.4 and W 33 ¼ 137.7 are found to be in close agreement with the corresponding values U 33 ¼ 68.67; V 33 ¼ 274.59 and W 33 ¼ 137.31 stated in Ref. [9]. Finally, we can check the results of our fully threedimensional numerical relaxation with those obtained by minimizing the two-dimensional, total energy functional of an axially symmetric Skyrme configuration.…”

“…Spin and isospin quantum numbers of ground states and excited states have so far almost exclusively been calculated within the rigid-body approach [6][7][8][9], that is by neglecting any deformations and symmetry changes due to centrifugal effects. Our numerical full field simulations clearly demonstrate the limitations of this simplification.…”

“…[9]. For comparison, the energy value M 1 ¼ 1.415 calculated in our full 3D simulation differs by 0.07% compared to the greater accuracy hedgehog ansatz (20).…”

“…Traditionally this is done by treating Skyrmions as rigid bodies that can rotate in both space and isospace. This means that one quantizes just the rotational and isorotational zero modes of each Skyrmion solution of a given B and then determines the spin and isospin quantum numbers [6][7][8][9] which are compatible with the symmetries of the static, classical soliton. This approach neglects any deformations and symmetry changes due to centrifugal effects.…”