Topological current of point defects and its bifurcation

TL;DR

This paper derives the topological current for point defects in 3D vector fields, linking defect charge to topological indices, and analyzes their evolution, especially during bifurcation when Jacobian vanishes.

Contribution

It introduces a topological current framework for point defects and explores their bifurcation behavior when Jacobian determinants are zero.

Findings

Defects' charge is determined by Hopf indices and Brouwer degrees.

Existence of branch processes during defect evolution when Jacobian equals zero.

Provides a mathematical description of defect bifurcation phenomena.

Abstract

From the topological properties of a three dimensional vector order parameter, the topological current of point defects is obtained. One shows that the charge of point defects is determined by Hopf indices and Brouwer degrees. The evolution of point defects is also studied. One concludes that there exist crucial cases of branch processes in the evolution of point defects when the Jacobian $D(\frac \phi x)=0$.