Self-Dual Vortices in the Fractional Quantum Hall System

TL;DR

This paper derives an exact self-dual equation with a topological term for the fractional quantum Hall system, revealing the existence and detailed topological structure of self-dual vortices, including their quantized charges and dynamic interactions.

Contribution

It introduces a new exact self-dual equation incorporating a topological term and analyzes the topological structure and dynamics of vortices in the fractional quantum Hall system.

Findings

Existence of self-dual vortices in the fractional quantum Hall system.

Vortex topological charges are quantized by Hopf indices and Brouwer degrees.

Vortices undergo generation, annihilation, splitting, and merging at critical points.

Abstract

Based on the $\phi$-mapping theory, we obtain an exact Bogomol'nyi self-dual equation with a topological term, which is ignored in traditional self-dual equation, in the fractional quantum Hall system. It is revealed that there exist self-dual vortices in the system. We investigate the inner topological structure of the self-dual vortices and show that the topological charges of the vortices are quantized by Hopf indices and Brouwer degrees. Furthermore, we study the branch processes in detail. The vortices are found generating or annihilating at the limit points and encountering, splitting or merging at the bifurcation points of the vector field $\vec\phi$.