2D Skyrmion topological charge of spin textures with arbitrary boundary conditions: a two-component spinorial BEC as a case study

TL;DR

This paper derives a comprehensive formula for calculating the topological charge of 2D spin textures, applicable to arbitrary boundary conditions, and demonstrates its application to spinor Bose-Einstein condensates with magnetic field-induced Skyrmions.

Contribution

It introduces a general expression for Skyrmion topological charge that accounts for arbitrary boundary conditions and relates the charge to local spin texture properties, extending previous formulas.

Findings

Derived a general Skyrmion charge formula for arbitrary boundaries.

Connected Skyrmion charge to local spin texture and singularities.

Applied the theory to 2D spinor BECs with magnetic field effects.

Abstract

We derive the most general expression for the Skyrmion topological charge for a two-dimensional spin texture, valid for any type of boundary conditions or for any arbitrary spatial region within the texture. It reduces to the usual one $Q = 1/4\pi \iint \vec f \cdot \left(\partial_x \vec f \times \partial_y \vec f\right)$ for the appropriate boundary conditions, with $\vec f$ the spin texture. The general expression resembles the Gauss-Bonet theorem for the Euler-Poincar\'e characteristic of a 2D surface, but it has definite differences, responsible for the assignment of the proper signs of the Skyrmion charges. Additionally, we show that the charge of a single Skyrmion is the product of the value of the normal component of the spin texture at the singularity times the Index or winding number of the transverse texture, a generalization of a Poincar\'e theorem. We illustrate our general results analyzing in detail a two-component spinor Bose-Einstein condensate in 2D in the presence of an external magnetic field, via the Gross-Pitaevskii equation. The condensate spin textures present Skyrmions singularities at the spatial locations where the transverse magnetic field vanishes. We show that the ensuing superfluid vortices and Skyrmions have the same value for their corresponding topological charges, in turn due to the structure of the magnetic field.