See the published version in Phys. Rev. B

Two-component superconductors are described by a doublet $\Psi=(\psi_1,\psi_2)^{\mathrm{T}}$ of complex fields $\psi_a=|\psi_a|\mathrm{e}^{i\varphi_a}$ (with $a=1,2$), whose squared moduli $|\psi_a|^2$ represent the density of individual superconducting components. Each of the components is coupled to the vector potential $\A$ of the magnetic field $\B=\Curl\A$, via the gauge derivative $\D=\Grad+ig\A$. Such a system is described by the Ginzburg-Landau free energy $E=\int {\cal E} d{\bf r}$, whose density reads as: \begin{align*} {\cal E}&= \frac{\B^2}{2}+\sum_{a=1,2}\frac{\gamma_a}{2}|\D\psi_a|^2 +\sum_{a,b=1,2}\frac{\mu_{ab}}{2} {\bs J}_a\cdot{\bs J}_b +{\nu}\left(\Psi^\dagger\Psi-1\right)^2+V[\Psi,\Psi^\dagger] \,, \\ &\text{where}~~{\bs J}_a=\Im(\psi_a^*\D\psi_a)=|\psi_a|^2(\Grad\varphi_a+g\A) \,. \end{align*} The terms $\mu_{12}=\mu_{21}$ of the current coupling matrix $\hat{\mu}$ describe the intercomponent drag. The free energy is supplemented with additional terms that explicitly break the global $\mathrm{SU}(2)$ symmetry of $U$ down to different subgroups \begin{equation} V[\Psi,\Psi^\dagger] = \sigma|\psi_1|^2|\psi_2|^2 +\eta\left(\psi_1\psi_2^*+\psi_1^*\psi_2\right) \,. \end{equation}

Numerical methods and details for simulations: To model

$\mathrm{U}{(1)}\!\times\!\mathrm{U}{(1)}$ superconductor, where the parameters $\gamma_a=0.02$ and $\mu_{ab}=1$

The displayed quantities are : Cyan and magenta tubes denote the positions of cores of (fractional) vortices defined as lines for which the densities of component $|\psi_1|^2$ or $|\psi_2|^2$ vanish. Tubes here are density-isosurfaces such that $|\psi_{1,2}|^2=2.5\times10^{-2}$.

1. F. N. Rybakov, J. Garaud and E. Babaev,
   Stable Hopf-Skyrme topological excitations in the superconducting state.
   Phys. Rev. B. 100, 094515 (2019). Link , arXiv