L / (N_\mathrm{J} + 1)$ is the distance between phase jumps of $\phi_\mathrm{J}$. If the phase jumps do not affect the Majorana modes, the regions of $\mu$ with vanishing $\epsilon^\pm$ should not change, and when they vanish, the corresponding eigenmodes should occupy the ends of the system in spite of the internal phase jumps.

![(a) Grayscale plots of $\epsilon^+$ for $g=\sqrt5-2$, $\hat{{\mbox{\boldmath $B$}}}=\hat{{\mbox{\boldmath $z$}}}$, $L=200$, $(\Gamma, \alpha, V_\mathrm{Q}) = (0.3, 0.3, 0.5)$, $\Delta = 0.1$, $-0.6\leq\mu\leq1$, $N_\mathrm{J}$ and $0 \leq \phi_\mathrm{J} \leq \pi$. (b) Value of $\epsilon^+$ plotted against $\mu$ for $N_\mathrm{J} = 1, 3, 7$ and $\phi_\mathrm{J} = \pi$. Eigenstates for $\mu = 0.04$ are plotted for (c) $N_\mathrm{J} = 1$, (d) $N_\mathrm{J} = 3$, and (e) $N_\mathrm{J} = 7$. []{data-label="fig:phiJ"}](Full_Fig7_200){width="8.66cm"}

In Fig. \[fig:phiJ\] we plot the value of $\epsilon^+$ for (a) a single jump with different sizes of phase jump $\phi_\mathrm{J}$, and (b) different numbers of phase jumps with $\phi_\mathrm{J} = \pi$. In Fig. \[fig:phiJ\] (a), $\phi_\mathrm{J} = 0$ corresponds to a system without a phase jump. Introduction of a single phase jump almost does not change the locations of the regions with vanishing $\epsilon^\pm$, though we observe a few curves of kinks in $\epsilon^+$ corresponding to a 