Discrete Skyrmions in 2+1 and 3+1 Dimensions

This paper describes a lattice version of the Skyrme model in 2+1 and 3+1 dimensions. The discrete model is derived from a consistent discretization of the radial continuum problem. Subsequently, the existence and stability of the skyrmion solutions existing on the lattice are investigated. One consequence of the proposed models is that the corresponding discrete skyrmions have a high degree of stability, similar to their continuum counterparts.

💡 Research Summary

The paper presents a systematic discretization of the Skyrme model in both (2+1)‑dimensional “baby” form and the full (3+1)‑dimensional SU(N) version, with a focus on preserving the topological stability that characterises the continuum theory. Starting from the continuum Lagrangian, which consists of the O(3) sigma‑model term, a Skyrme term that prevents scale collapse, and a potential term fixing the vacuum, the authors impose radial symmetry. In the baby Skyrme case the field is written as a hedgehog configuration φ_i = k_i sin g(r,t), φ_3 = cos g(r,t), where k_i encodes the azimuthal angle and g(r,t) is a real profile function. The topological charge becomes the winding number of the map S²→S².

To pass to a lattice, the radial coordinate r is replaced by discrete points r_n = n h (h is the lattice spacing). Instead of a simple finite‑difference, the authors adopt a nonlinear “sine‑difference” scheme inspired by previous work on the O(3) sigma model: derivatives are approximated by expressions involving sin