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
  1. J. Garaud and E. Babaev,
    Domain walls and their experimental signatures in s+is superconductors.
    Phys. Rev. Lett. 112, 017003, (2014). Link , arXiv