Existence of global vortices in a class of Ginzburg-Landau models

TL;DR

This paper investigates the existence and properties of global vortices in a two-component Ginzburg-Landau model, analyzing different physical states through mathematical methods.

Contribution

It provides a rigorous mathematical analysis of vortex solutions in the TCGL model, including existence, uniqueness, and asymptotic behavior for different vacuum states.

Findings

Existence of vortex solutions in both 1VEV and 2VEV cases.

Unique solutions with specific asymptotic properties.

Quantization of solutions related to physical states.

Abstract

In this paper, the two-component Ginzburg-Landau (TCGL) model, which is an important research object in mathematics and physics of two scalar fields with $U(1)\times U(1)$ symmetry, is studied in detail by the shooting method and the fixed point theorem, which contains two cases: single vacuum expectation value (1VEV) and two vacuum expectation values (2VEV). This represents two different physical states: superconductivity-normal state and superconductivity superconductivity state, and we prove the existence, uniqueness, asymptotic properties and quantization of the solutions to these two boundary value problems.