in the following.

Bogoliubov-de Gennes equation: zero modes and Majorana fermions
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The Hamiltonian (\[eqn:Hamiltonian\_gen\]) is bilinear in operators $\hat{c}$ and $\hat{c}^\dag$. It can be exactly diagonalized in the Nambu spinor space $\{ ({\mbox{\boldmath $u$}}_\uparrow, {\mbox{\boldmath $u$}}_\downarrow, {\mbox{\boldmath $v$}}_\uparrow, {\mbox{\boldmath $v$}}_\downarrow)^\mathrm{T} \}$, with the basis obtained as the set of the eigenvectors of the Bogoliubov-de Gennes (BdG) equation.

Alternatively, fermion pairing via a short-range attractive interaction can also be simulated by introducing a pairing constant $g$ and solving the Bogoliubov-de Gennes equation self-consistently. In a lattice system the energy cutoff which renormalizes $g$ can be in principle determined from the lattice constant. However, here we fix the value of $\Delta$, a homogeneous proximity pairing, by hand. This is because we do not expect that the BdG approximation, which simulates the pair formation *within* the 1D wire, directly corresponds to our Hamiltonian (\[eqn:Hamiltonian\_gen\]).

We solve [@Iskin2012; @Liu2012] $$\left(\begin{array}{cccc}
\hat{H_{\uparrow\uparrow}} & \hat{H_{\uparrow\downarrow}} & 0 & \Delta\\
\hat{H_{\downarrow\uparrow}} & \hat{H_{\downarrow\downarrow}} & -\Delta & 0\\
0 & -\Delta & -\hat{H_{\uparrow\uparrow}} & -\hat{H_{\uparrow\downarrow}}\\
\Delta & 0 & -\hat{H_{\downarrow\uparrow}} & -\hat{H_{\downarrow\downarrow}}\\
\end{array}\right)
\Psi = \epsilon \Psi,
\label{eqn:BdG}$$ in which $\Psi = ({\mbox{\boldmath $u$}}_\uparrow, {\mbox{\boldmath $u$}}_\downarrow, {\mbox{\boldmath $v$}}_\uparrow, {\mbox{\boldmath $v$}}_\downarrow)^\mathrm{T}$ is a $4L$-dimensional vector and $\hat{H_{\sigma\sigma'}}$ are 