Lattice Three-Dimensional Skyrmions Revisited

TL;DR

This paper constructs and analyzes lattice analogues of three-dimensional skyrmions, demonstrating their stability, bifurcation structure, and continuum limit behavior through numerical methods and stability analysis.

Contribution

It introduces a method to create topologically stable lattice skyrmions using the Skyrme model and analyzes their stability and bifurcation properties.

Findings

Stable lattice skyrmion solutions are identified.

Bifurcation structure near the continuum limit is characterized.

Spectrally stable solutions are confirmed via numerical simulations.

Abstract

In the continuum a skyrmion is a topological nontrivial map between Riemannian manifolds, and a stationary point of a particular energy functional. This paper describes lattice analogues of the aforementioned skyrmions, namely a natural way of using the topological properties of the three-dimensional continuum Skyrme model to achieve topological stability on the lattice. In particular, using fixed point iterations, numerically exact lattice skyrmions are constructed; and their stability under small perturbations is explored by means of linear stability analysis. While stable branches of such solutions are identified, it is also shown that they possess a particularly delicate bifurcation structure, especially so in the vicinity of the continuum limit. The corresponding bifurcation diagram is elucidated and a prescription for selecting the branch asymptoting to the well-known continuum limit is given. Finally, the robustness of the spectrally stable solutions is corroborated by virtue of direct numerical simulations .