Two-orbital fermionic pairing Hamiltonian ($H=\epsilon (a^{†}a+b^{†}b)+\Delta (ab+b^{†}{a}^{†})$)

For distinct fermionic orbitals obeying the full canonical anticommutation relations, fix the Fock state order $|11⟩=a^{†}{b}^{†}|0⟩$. With real $\epsilon ,\Delta >0$, the even-fermion parity matrix is $\left(\begin{array}{cc}0& -\Delta \\ -\Delta & 2\epsilon \end{array}\right)$, since $ab|11⟩=-|0⟩$. The two odd-fermion parity energy eigenvalues are both $\epsilon$. The even-fermion parity energy eigenvalues are $\epsilon \pm \sqrt{{\epsilon }^2+{\Delta }^2}$. This finite model isolates the relative sign and constant shift required in a fermionic Bogoliubov transformation without any momentum sum or particle-hole doubling factor.