A new topological aspect of the arbitrary dimensional topological defects

TL;DR

This paper introduces a new topological current framework to describe arbitrary dimensional topological defects, revealing their generation, quantization, bifurcation, and interactions based on the order parameter field and the $ ext{phi}$-mapping method.

Contribution

It develops a generalized topological current approach for arbitrary dimensional defects, including their creation, quantization, bifurcation, and interactions, using the $ ext{phi}$-mapping method.

Findings

Topological defects originate from zero points of the order parameter field.

Topological charges are quantized via Hopf indices and Brouwer degrees.

Defects can split or merge at degenerate points while conserving total charge.

Abstract

We present a new generalized topological current in terms of the order parameter field $\vec \phi$ to describe the arbitrary dimensional topological defects. By virtue of the $% \phi$-mapping method, we show that the topological defects are generated from the zero points of the order parameter field $\vec \phi$, and the topological charges of these topological defects are topological quantized in terms of the Hopf indices and Brouwer degrees of $\phi$-mapping under the condition that the Jacobian $% J(\frac \phi v)\neq 0$. When $J(\frac \phi v)=0$, it is shown that there exist the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, we detail the bifurcation of generalized topological current and find different directions of the bifurcation. The arbitrary dimensional topological defects are found splitting or merging at the degenerate point of field function $\vec \phi $ but the total charge of the topological defects is still unchanged.