of $\kappa$. Particularly, even for values of $\kappa$ such as $\pi/3$ and $\pi/2$, $\epsilon^+$ vanishes when the chemical potential lies in the region where single particle eigenstates are effectively spinless.

In these regions, because of the effectively single band structure of the non-interacting Hamiltonian, the Kitaev model is realized, with Majorana fermion modes localizing at system ends. [@Tezuka2012] The value of $\epsilon^+$ becomes closer to $0$ as $L$ is increased, reflecting that the end modes are more separated. For smaller $L$, the separation between the single particle eigenstates is comparable to $\Delta$ when $\Delta\ll\{\Gamma, t\}$. In this case, strength of the induced superconductivity depends on the relative location of a level and the chemical potential. When the separation is larger (the density of states is lower) the value of $\epsilon^+$ usually stays larger, but changes rapidly. $\epsilon^+$ becomes more homogeneous and generally reduced as $\Delta$ is increased inside each sub-band which does not overlap in energy with another.

We note that, while $N_\downarrow$ is always larger than $N_\uparrow$ for $\Gamma > 0$, there are sub-bands having fermions almost polarized in the $+\hat{{\mbox{\boldmath $z$}}}$ direction ($\langle S_z \rangle \sim 1/2$) even when $\mu < 1$ with less than half filling factor 