the localization of the Majorana operators observed by DMRG [@Tezuka2012].

We have plotted the values of $\epsilon^\pm$ as a function of $\mu$ along with the single particle state energy in Fig. \[fig:localization\] (a–b). In Fig. \[fig:localization\] (a) with $V_\mathrm{Q} = 0$, the overlapping RZ bands are clearly observed, and $\epsilon^\pm$ vanish only when the chemical potential crosses only one of the RZ bands.

In Fig. \[fig:localization\] (b) with $V_\mathrm{Q} = 0.5$ several regions with vanishing $\epsilon^\pm$ are found, each corresponding to the energy region with states from just one of the two Hofstadter-butterfly-like structure. The components of the eigenvector corresponding to the eigenvalue $\epsilon^+$ are shown in Fig. \[fig:localization\] (c–e) for values of $\mu$ approaching to one of the regions with vanishing $\epsilon^\pm$. The localization of the mode is clearly observed with an increase of $\sqrt{\langle x^2 \rangle}$. In all regions with vanishing $\epsilon^\pm$, we observe a clear localization of the corresponding eigenvectors of $M_\mathrm{BdG}$.

![(Color online) Top figure: Single particle eigenstates of eq. (\[eqn:Hamiltonian\_gen\]) color-coded according to the value of $S_z$ for $\hat{{\mbox{\boldmath $B$}}}=\hat{{\mbox{\boldmath $z$}}}$, $L=400$, $(\Gamma, \alpha, V_\mathrm{Q}) = (0.3, 0.3, 0.2)$ and $-0.6\leq\mu\leq1$, $0 < \kappa < \pi$. Lower figures: Grayscale plots of $\epsilon^+$ for the same 