Oral presentation

Session Information

Location: Science and Cultural Center - FORTH
Day: 2. Thursday 17th
Time: 11:30-11:50
Chairperson: to be announced…

Presentation Details

Presentation Type: Oral presentation
Title: Towards a Fractional Theory of Chiral Skyrmions
Abstract: We propose a fractional extension of the classical theory of chiral skyrmions by introducing nonlocal and memory effects into the underlying micromagnetic model. In the standard micromagnetic framework, the energy functional combines the ferromagnetic exchange interaction, represented by the Dirichlet energy, the antisymmetric Dzyaloshinskii–Moriya (DM) interaction responsible for chirality, and magnetic anisotropy. The main objective is to investigate which of these physical interactions should be modeled as nonlocal. A natural starting point is to replace the classical exchange interaction by a fractional one, leading to a nonlocal micromagnetic energy that combines fractional exchange with local DM interaction and anisotropy. The fractional Laplacian provides a mathematically consistent description of long-range exchange interactions while establishing a direct connection with the theory of fractional harmonic maps. To account for heterogeneous media, we further propose a variable-order fractional Laplacian, whose order varies spatially according to material properties, defects, or interfaces. Such a model offers a natural framework for describing spatially varying nonlocal interactions and may provide a new mathematical approach to skyrmion pinning phenomena. Another promising direction is the introduction of a fractional DM interaction through an appropriate fractional curl operator that preserves chirality. Memory effects in magnetization dynamics can be incorporated by replacing the classical time derivative in the Landau–Lifshitz–Gilbert equation with a Caputo fractional derivative, resulting in a nonlinear constrained evolution equation with temporal memory. A unified space-time fractional model would combine long-range spatial interactions with memory effects. These proposed models open a wide range of challenging problems in partial differential equations and the calculus of variations, including the existence and regularity of minimizers, stability of skyrmions, topological constraints, fractional Sobolev formulations, uniqueness of radial solutions, and the asymptotic behavior as the fractional order approaches the classical limit. A further long-term objective is to develop a fractional analogue of the skyrmion number by replacing classical derivatives with suitable fractional differential operators while preserving topological invariance. Such a construction would provide a natural link between fractional calculus and the topology underlying magnetic textures.

Presenter

Prof Ljubica Oparnica
Faculty of Education in Sombor, University of Novi Sad | Serbia

Authors

1. Oparnica, Ljubica | Faculty of Education in Sombor, University of Novi Sad