  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.
