Topological invariants of vortices, merons, skyrmions, and their combinations in continuous and discrete systems

TL;DR

This paper develops a unified topological classification framework for magnetic vortices, merons, and skyrmions, applicable to both continuous and discrete systems, using homotopy group analysis to derive invariants.

Contribution

It introduces a comprehensive topological classification method that unifies the treatment of vortices and skyrmions across different physical systems.

Findings

Derives topological invariants forming a $\

Provides explicit calculation methods for invariants in various systems

Extends the classification framework beyond magnetism to general physical systems.

Abstract

Magnetic vortices and skyrmions are typically characterized by distinct topological invariants. This work presents a unified approach for the topological classification of these textures, encompassing isolated objects and configurations where skyrmions and vortices coexist. Using homotopy group analysis, we derive topological invariants that form the free abelian group, $\mathbb{Z}\times\mathbb{Z}$. We provide an explicit method for calculating the corresponding integer indices in continuous and discrete systems. This unified classification framework extends beyond magnetism and is applicable to physical systems in general.