Characterizing topological properties in 2D textures with magnetic structure tensors

The spin structures, including magnetic skyrmions, have attracted great attention due to their interesting properties caused by their topology. We introduce a magnetic structure tensor as a powerful tool for characterizing topological properties in two-dimensional spin textures. We derived an order parameter from the two orthogonal components of the magnetic structure tensor. The order parameter effectively analyzes the structures of the spin patterns and identifies topological defects such as the end or branching points of linear chiral structures and magnetic skyrmions, as a pole or multipoles composed of topological quantized charges. We present several spin configurations interpreted as topological monopoles, dipoles, and quadrupoles formed by the combination of these topological poles. This approach provides valuable insights into defect distribution and interactions, validated through both simulations and experimental data. Moreover, the approach can be generally applied for effective analysis of self-organized patterns such as fingerprints.