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A Bogoliubov--de Gennes Hamiltonian is the matrix representation of a quadratic fermion Hamiltonian in a Nambu spinor combining annihilation operators and creation operators. It treats ordinary one-particle motion and fermionic pairing in one eigenvalue problem.

For a column $c$ of $N$ fermionic annihilation operators, let $h=h^{†}$ and $\Delta =-{\Delta }^T$. With $\Psi =(c,c^{†}{}^T{)}^T$, defineThe canonical anticommutation relations give the identityThe factor $1/2$ compensates for the full particle-hole doubling; the trace constant follows when the hole block is normal-ordered.

A two-by-two opposite-spin pairing blockhas eigenvalues $E_{\pm }=\pm \sqrt{{\xi }^2+|\Delta {|}^2}$, as follows from its characteristic polynomial. Pairing opens a spectral gap at $\xi =0$. Negative and positive partners in a fully doubled representation are related by Particle-hole symmetry of a Bogoliubov--de Gennes Hamiltonian; they are not independent species of physical quasiparticles. A reduced block and a full doubled basis therefore require consistent counting conventions.

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.

Position-space particle-hole symmetry complex-conjugates plane waves, so the Fourier transform sends $k$ to $-k$. Fourier transforming the stated relation therefore givesIf $H_{\mathrm{BdG}}(k)u_k={\epsilon }_k{u}_k$, complex conjugation and multiplication by ${\Sigma }_1$ yieldThus the Particle-hole symmetry of a Bogoliubov--de Gennes Hamiltonian imposesor, band by band after a suitable relabelling, ${\epsilon }_{-k}=-{\epsilon }_k$.