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For real gap and the particle-hole block $\left(\begin{array}{cc}{\xi }_k& -{\Delta }_k\\ -{\Delta }_k& -{\xi }_k\end{array}\right)$, the BCS coherence factors obey $u_k^2=(1+{\xi }_k/E_k)/2$, $v_k^2=(1-{\xi }_k/E_k)/2$ and $u_k{v}_k={\Delta }_k/(2E_k)$, where $E_k=\sqrt{{\xi }_k^2+{\Delta }_k^2}$. They diagonalize the Bogoliubov--de Gennes Hamiltonian and give the empty/pair amplitudes of the BCS ground state. Their relative sign depends on the pairing convention.

For real ${\Delta }_k$ and ${\xi }_{-k}={\xi }_k$, introduce the particle-hole column ${\Psi }_k=(c_{k\uparrow },c_{-k\downarrow }^{†}{)}^T$. Its quadratic block is the Bogoliubov--de Gennes HamiltonianThe BCS coherence factors can be chosen to obeyWith $u_k\ge 0$, choose the sign of $v_k$ to match ${\Delta }_k$. The rotation diagonalizes the matrix to $\mathrm{diag}(E_k,-E_k)$. Reordering ${\gamma }_{-k\downarrow }{\gamma }_{-k\downarrow }^{†}=1-{\gamma }_{-k\downarrow }^{†}{\gamma }_{-k\downarrow }$ and restoring the mean-field constant givesThe inverse transformation is ${\gamma }_{k\uparrow }=u_k{c}_{k\uparrow }-v_k{c}_{-k\downarrow }^{†}$ and ${\gamma }_{-k\downarrow }=v_k{c}_{k\uparrow }^{†}+u_k{c}_{-k\downarrow }$. Both annihilate $(u_k+v_k{b}_k^{†})|0⟩$. Since every positive-energy quasiparticle mode is empty, the normalized BCS ground state isEach momentum label refers to the pair $(k\uparrow ,-k\downarrow )$ once; distinct labels use disjoint single-particle spin states.

The continuum Hamiltonian is that of a spinless one-dimensional p-wave topological superconductor. Its topological phase has $\mu >0$, whereas the empty vacuum may be modeled as the trivial phase with $\mu <0$. With superconductor at $x<0$ and vacuum at $x>0$, this is

Near the transition $\mu =0$, momenta are small and the quadratic term may be neglected. In Pauli-matrix notation, the long-wavelength Bogoliubov--de Gennes Hamiltonian isThe zero-energy equation becomesChoose a constant spinor ${\chi }_{+}$ with ${\tau }_y{\chi }_{+}={\chi }_{+}$. Thendecays on both sides because $\mu$ changes from positive to negative. For asymptotically constant $|\mu |$, the Continuum p-wave Majorana interface mode has widthIts characteristic momentum is $|p|\sim |\mu |/\Delta$. The neglected kinetic energy is small compared with $|\mu |$ when $|\mu |\ll m{\Delta }^2$ up to a factor of two, equivalently when $\xi \gg \hslash /(m\Delta )$.

In the Nambu basis, Particle-hole symmetry of a Bogoliubov--de Gennes Hamiltonian lets a zero-energy eigenvector be chosen self-conjugate, $v(x)=u(x{)}^{\ast }$. The corresponding quasiparticle operator isTaking the adjoint and interchanging the two terms gives ${\gamma }^{†}=\gamma$. With the usual normalization it obeys $\{\gamma ,\gamma \}=2$, so it is a Majorana fermion operator localized at the interface.

The Bogoliubov--de Gennes Hamiltonian has particle-hole symmetry. A locally nondegenerate zero mode can therefore be chosen particle-hole invariant. If its Nambu wavefunction is $(u_j,u_j^{\ast })$, defineThen ${\gamma }_j^{†}={\gamma }_j$. Exponential localization and $\ell /\xi \to \infty$ make distinct zero-mode wavefunctions orthogonal. Normalizing each one and using the fermionic canonical anticommutation relations gives

Pair the $2M$ Majorana zero modes into ordinary zero-energy fermionsTheir occupations produce $2^M$ states. Physical operations preserve total fermion parity, so choosing one parity sector imposes one binary constraint and leaves $2^{M-1}$ states. Thus Dense encoding with Majorana zero modes gives

Two local gapped Hamiltonians are topologically equivalent when a continuous path of local Hamiltonians joins them without closing the bulk gap. The Bogoliubov--de Gennes Hamiltonian has energiesFor $t,\Delta >0$, this gap can close only atThe whole region $-2t<\mu <2t$ is connected and gapped, so its parameters can be continuously deformed to $t=\Delta ,\mu =0$. Therefore

Writingshows the topological distinction. As $k$ crosses the Brillouin zone, $(d_y,d_z)$ traces an ellipse. It encloses the origin once when $|\mu |<2t$, giving nonzero winding number of a one-dimensional Bogoliubov--de Gennes Hamiltonian. For $\mu <-2t$ it does not enclose the origin and has winding zero. Changing this integer requires the ellipse to pass through the origin, exactly the bulk gap closing.