Investigation of the stability of Hopfions in the two-component Ginzburg-Landau model

TL;DR

This paper investigates the stability of Hopfions in a two-component Ginzburg-Landau model, combining analytical and numerical methods, and finds that Hopfions tend to shrink into singular configurations, challenging their stability.

Contribution

It provides a comprehensive analysis of Hopfion stability in the two-component Ginzburg-Landau model using both analytical and numerical approaches, including various potentials.

Findings

Hopfions tend to shrink into thin loops

Stability is compromised by singular configurations

Numerical methods cannot fully resolve the final state

Abstract

We study the stability of Hopfions embedded in the Ginzburg-Landau (GL) model of two oppositely charged components. It has been shown by Babaev et al. [Phys. Rev. B 65, 100512 (2002)] that this model contains the Faddeev-Skyrme (FS) model, which is known to have topologically stable configurations with a given Hopf charge, the so-called Hopfions. Hopfions are typically formed from a unit-vector field that points to a fixed direction at spatial infinity and locally forms a knot with a soft core. The GL model, however, contains extra fields beyond the unit-vector field of the FS model and this can in principle change the fate of topologically non-trivial configurations. We investigate the stability of Hopfions in the two-component GL model both analytically (scaling) and numerically (first order dissipative dynamics). A number of initial states with different Hopf charges are studied; we also consider various different scalar potentials, including a singular one. In all the cases studied, we find that the Hopfions tend to shrink into a thin loop that is too close to a singular configuration for our numerical methods to investigate.