This is a supplementary material for
arxiv:1308.3220,

This is a supplementary material for
arxiv:1308.3220.
See also supplementary information
Here.
See the published version in Phys. Rev. Lett.

Introduction

**The three-component Ginzburg-Landau**free energy is \begin{align*} {\cal F}= \frac{\B^2}{2} +&\sum_{a=1}^3\frac{1}{2}|\D\psi_a|^2 + \alpha_a|\psi_a|^2+\frac{1}{2}\beta_a|\psi_a|^4 \\ -&\sum_{a=1}^3\sum_{b>a}^3 \eta_{ab}|\psi_a||\psi_b|\cos(\varphi_b-\varphi_a) \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 $\lambda=\frac{1}{e\sqrt{\sum_a|\psi_a|^2}}$.

To investigate the magnetization process of a three-band superconductor, in the type-II regime, 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 nonlinear conjugate gradient algorithm within a finite element formulation provided by Freefem++ library.

**Field cooled experiments :**There, the Gibbs energy is minimized at $T=T_c+\delta T$, for a given applied field $H$ exceeding the second critical field $H>H_{c2}$. The Gibbs energy is subsequently minimized for decreasing temperatures.

**Magnetization process at fixed $ T $ :**At a temperature $T<\Tz$, no field is initially applied ($H=0$) and the superconductor is in a uniform Meissner state. Note that in the presence of geometrically stabilized domain-wall, even at $H=0$ the system is not in a uniform Meissner state. Keeping the temperature fixed, the applied field is increased with a step $\delta H$.

0. Time-Reversal Symmetry Breaking

**The displayed quantity is :**This shows the minimum of the potential energy, as a function of the the two phase differences $ \varphi_2-\varphi_1 $ and $ \varphi_3-\varphi_1 $. Potential energy is thus minimized with respect to densities for a given phase configuration. The temperature dependence of the parameters is referred in the paper as

*Set-I*. This is shown for decreasing temperatures, past $T<\Tz\simeq 0.92$ and the temperature is given in units of $ T_c$.

**Movie 0 -**This shows the potential energy as a function of temperature. For temperatures above $ \Tz $, the ground state is unique and the Time Reversal Symmetry is unbroken. For temperatures below $ \Tz $, the ground state has discrete degeneracy and the Time Reversal Symmetry is broken.

1. Zero field cooled experiment - Geometric stabilization

**The displayed quantities are :**On the first line, the magnetic field $\B$ and the phase differences $ \varphi_2-\varphi_1 $ and $ \varphi_3-\varphi_1 $. The second line shows the densities of the superconducting condensates $ |\psi_1|^2 $, $ |\psi_2|^2 $ and $ |\psi_3|^2 $ respectively. The temperature is given in the units of $ T_c $.

**Movie 1 -**This shows a three-band superconductor being cooled through the Broken Time-Reversal Symmetry transition, in zero external field. At temperatures higher than $\Tz$, the system is uniform Meissner state. At the symmetry breaking temperature $\Tz$, in different regions the system assumes two different phase configurations. This leads to the formation of a domain wall. The domain-wall is stabilized by the non-convex geomtry of the sample.

**Movie 2 -**This shows a three-band superconductor being cooled through the Broken Time-Reversal Symmetry transition, in zero external field. At temperatures higher than $\Tz$, the system is uniform Meissner state. At the symmetry breaking temperature $\Tz$, in different regions the system assumes two different phase configurations. This leads to the formation of a domain wall. Here, the domain wall is stabilized by pinning defects.

2. Field cooled experiment

**The displayed quantities are :**On the first line, the magnetic field $\B$ and the phase differences $ \varphi_2-\varphi_1 $ and $ \varphi_3-\varphi_1 $. The second line shows the densities of the superconducting condensates $ |\psi_1|^2 $, $ |\psi_2|^2 $ and $ |\psi_3|^2 $ respectively. The temperature is given in the units of $ T_c $.

**Movie 3 -**This shows a three-band superconductor being cooled through the Broken Time-Reversal Symmetry transition, in external field $ HS/\Phi_0=70 $. At temperatures higher than $\Tz$, the system forms simple vortex lattice. At the symmetry breaking temperature $\Tz$, in different regions the system assumes two different phase configurations. This leads to the formation of a domain-wall, which cannot freely decay. The preexisting vortices stabilize the domain-wall.

3. Magnetization process at fixed T

**The displayed quantities are :**On the first line, the magnetic field $\B$ and the phase differences $ \varphi_2-\varphi_1 $ and $ \varphi_3-\varphi_1 $. The second line shows the densities of the superconducting condensates $ |\psi_1|^2 $, $ |\psi_2|^2 $ and $ |\psi_3|^2 $ respectively. $ N $ is the number of applied flux quanta. The corresponding applied field on a sample of surface $ S $, is $ H=N\Phi_0/S $.

**Movie 4 -**This shows the magnetization process of the same system as in Movie 2. That is starting at temperature below $\Tz$. The field is then slowly increased. Because of Bean-Livingston barrier, vortices enter for applied field above $ H_{c1} $. At the final stages, the step in the applied field is big enouh to locally push the system to another phase locking, thus nucleating a domain wall. The later is stabilized by the vortices.

**Movie 5 -**This shows the reference magnetization process of the same system as in Movie 1. That is starting at temperature below $\Tz$, when a domain wall has been geometrically stabilized by the non-convex geometry. The field is then slowly increased. Because of depleted densities on the domain-wall, (fractional) vortices enter on the domain-wall. Vortices enter for applied field way below $ H_{c1} $, since on the domain wall the first critical field is effectively smaller.

**Movie 6 -**This shows the magnetization process of the same system as in Movie 3. That is starting at temperature below $\Tz$, when no domain wall has been geometrically stabilized by the non-convex geometry. The field is then slowly increased. Because of Bean-Livingston barrier, vortices enter for applied field above $ H_{c1} $.

References