Theoretical Studies of 2-Dimensional Skyrmion Lattices

Abstract

As a unified field theory of mesons and baryons in 3+1 dimensions, the Skyrme Model has proven to represent an interesting formulation to considering nucleons, with crucial connections to Quantum Chromodynamics. The study of cubic arrays of Skyrme solitons (or "skyrmions") in particular underscores a broad array of considerations regarding multi-skyrmion interactions, with key methods to considering potential branches for further investigation regarding these systems. One such direction for further research is the question of lattice structures in R^2, which have experienced a resurgence of interest in light of experimental verification of these 2-dimensional skyrmion lattices. As such, in this paper, we study the properties of square lattice structures of 2-dimensional skyrmions, specifically the cases of those using twisted or periodic boundary conditions. After reviewing the construction of the 2-Dimensional Skyrme model and its properties with regard to the hedgehog ansatz, we determine ansatz forms for the twisted and periodic square lattices as Fourier series which satisfy the overarching symmetries of the system. We then truncate our Fourier expansions to a finite number of terms, plug said expansion into the base energy functional, and determine the Fourier coefficients which minimize the energy per lattice unit while maintaining certain constraints regarding the topological degree over each lattice unit, reflective of the properties stemming from the choice of boundary conditions. From this procedure, we determine that, at various values of the mass-squared parameter µ^2, the square lattice with twisted boundary conditions demonstrates a lower energy per lattice unit than that with periodic boundary conditions. Furthermore, while the periodic square lattice yields a baryon density which converges to an array of full 2D skyrmions in the minimal-energy configuration, the twisted square lattice demonstrates a splitting of the minimal-energy baryon density into half-skyrmions, localized regions where the topological degree is approximately 1/2. We also extend our analysis to include additional terms in the energy functional, specifically an additional four-derivative term and that pertaining to the Dzyaloshinskii-Moriya interaction, whose strength we control via the corresponding coefficients κ and D, respectively. We consider various cases, including (a) κ ≠ 0 while D = 0, (b) κ = 0 while D ≠ 0, and (c) both κ, D ≠ 0. In our analysis of the Dzyaloshinskii-Moriya term in particular, we determine that, as the term coefficient D approaches a critical value D_crit from below (hence acting as an upper bound), the field configuration stretches along the boundaries of the square tessellation. We then finish this paper with some considerations regarding a lattice tessellated by equilateral triangles, and initial steps towards considering this topic.