(negative) eigenvalue, only $|v_{\sigma,l}|^2$ ($|u_{\sigma,l}|^2$) contributes to the particle distribution in the $T\rightarrow0$ limit.

![(a–b) Largest negative ($-$) and smallest positive ($+$) eigenvalues of the BdG equation (\[eqn:BdG\]) plotted against the chemical potential $\mu$ for $g = \sqrt{5}-2$, $(\Gamma, \alpha, \Delta)=(0.2, 0.3, 0.1)$ and $V_\mathrm{Q} = $(a) $0$, (b) $0.5$. (c–e) Amplitudes of the particle- and hole-like parts of the eigenvectors $\sum_\sigma|u_\sigma|^2$, $\sum_\sigma|v_\sigma|^2$ plotted against the lattice site for $(\Gamma, \alpha, \Delta)=(0.2, 0.3, 0.1)$ and $\mu = $(c) $-0.04$, (d) $-0.02$, (e) $-0.01$.[]{data-label="fig:localization"}](Full_Fig2_200){width="8.66cm"}

### Distribution of eigenvalues

For $L$ sites in the system, because of the spin and particle–hole degrees of freedom, we have $4L$ eigenstates of the BdG equation (\[eqn:BdG\]). The introduction of $\Delta$ opens a gap in the eigenvalue spectrum of eq. (\[eqn:BdG\]) in the absence of the spin-orbit coupling $\alpha$ or the Zeeman field $\Gamma$. With the spin-orbit coupling and the Zeeman field, when the chemical potential $\mu$ satisfies $$\mathrm{(lower\ band\ bottom)} < \mu < \mathrm{(upper\ band\ bottom)}$$ so that the Kitaev model [@Kitaev2001] is effectively realized in the spinful case, the system is a topological superconductor for $0 < \Delta \lesssim \Gamma$ when ${\mbox{\boldmath $B$}}$ is in the $z$–$x$ plane.[@Sau2010; @Alicea2010; @Lutchyn2010; @Oreg2010; @Stanescu1302.5433] In this 