= 0$ that when we fix the value of $\kappa$ one or more in-gap states emerge within the energy gaps due to the potential $\epsilon_{l}$. Such in-gap states also exist in our model with non-zero $\Gamma$ and $\alpha$, and can cross each other or another sub-band because the RZ bands are shifted in energy. As we change the value of the phase of the site energy modulation, $\phi_0$, the energies of these in-gap states change rapidly, while other states in ‘bulk’ sub-bands do not change significantly.

In Fig. \[fig:f-L200lambda0.5\] we change the value of $\phi_0$ to study the effect of such in-gap states. We have plotted the single particle state energies color-coded by the value of $\langle S_z \rangle$ as well as the value of $\epsilon^+$ for different system sizes. The plot at the bottom with $L = 200$ looks similar to that of the single particle state energies at the top, except that the regions with two overlapping sub-bands do not have a vanishing $\epsilon^+$.

We observe that the dependence of the eigenvalues on the choice of the phase $\phi_0$ becomes weaker as $L$ increases. Bulk sub-bands are not shifted, and crossing with a single in-gap state does not 