This is a supplementary material for
arxiv:1307.3211.
See the published version in
Phys. Rev. B .

Introduction

**The two-component Ginzburg-Landau**free energy is \begin{align*} {\cal F}= \frac{\B^2}{2} +&\sum_{a}\frac{1}{2}|\D\psi_a|^2 + \alpha_a|\psi_a|^2+\frac{1}{2}\beta_a|\psi_a|^4 \\ +&\frac{\nu}{2}\left|\Im(\psi_1^*\D\psi_1) +\Im(\psi_2^*\D\psi_2)\right|^2 \end{align*} with the covariant derivative $\D=\Grad+ie\A$ and the magnetic field $\B=\Curl\A$. The coupling constant $e$ parametrizes the London penetration length $\lambda=\frac{1}{e\sqrt{\sum_a|\psi_a|^2}}$.

**Description of the simulations :**To investigate the magnetization process of the Skyrmionic state, we simulate the Gibbs free energy ${ \cal G}={\cal F}-\B\cdot {\bf H} $ of the system, on a finite domain in an increasing external field $ {\bs H}=H{\bs e}_z $. Here, the energy is minimized using a quasi-Newton (BFGS) method within a finite difference scheme. The animations show the state of the system at a given value of the applied field.

1. Magnetization process : Ginzburg-Landau variables

**The displayed quantities are :**On the first column densities of the superconducting condensates $ |\psi_1|^2 $, $ |\psi_2|^2 $ and the magnetic field $\B$. On the second column are displayed the magnetization curve $ M(H) $, the phase difference $ \varphi_2-\varphi_1 $ and the total supercurrent $ \J$.

**Movie 1 -**This shows the Skyrmions magnetization process of a finite sample in a slowly increasing applied magnetic field. This corresponds to the regime displayed in Figure 5 of the paper.

2. Magnetization process : Pseudo-spin texture

The two-component model can be mapped to an easy-plane
non-linear $\sigma$-model. There, the pseudo-spin unit
vector $\bf n$ is the projection of superconducting condensates
on spin-$1/2$ Pauli matrices $\bs\sigma$:
$~{\bf n}=\frac{\Psi^\dagger\sigma\Psi}{\Psi^\dagger\Psi}$
where $\Psi^\dagger=(\psi_1^*,\psi_2^*)$. The pseudo-spin is
a map ${\bf n}:S^2\rightarrow S^2$, classified by the homotopy
class $\pi_2(S^2)\in \mathbb{Z}$

**The displayed quantities are :**the normalized projection of ${\bf n}$ onto the plane, while colors give the magnitude of $n_z$ . Blue correspond to the south pole (-1) while red is the north pole (+1) of the target sphere $S^2$.**Movie 2 -**This shows the magnetization process of the same system as in Movie 1.

**Movie 3 -**This shows the same magnetization process as in Movie 2, but zooming on the upper right corner. This gives better resolution of the Skyrmion entry in the sample, while the applied field is increased.

References