= Two-orbital fermionic pairing Hamiltonian
{title2=$H=\varepsilon(a^\dagger a+b^\dagger b)+\Delta(ab+b^\dagger a^\dagger)$}
= Two-mode fermion pairing Hamiltonian
{synonym}
For distinct <fermionic> orbitals obeying the full <canonical anticommutation relations>, fix the <Fock state> order $|11\rangle=a^\dagger b^\dagger|0\rangle$. With real $\varepsilon,\Delta>0$, the even-<fermion parity> <matrix> is $\begin{pmatrix}0&-\Delta\\-\Delta&2\varepsilon\end{pmatrix}$, since $ab|11\rangle=-|0\rangle$. The two odd-<fermion parity> <energy eigenvalues> are both $\varepsilon$. The even-<fermion parity> <energy eigenvalues> are $\varepsilon\pm\sqrt{\varepsilon^2+\Delta^2}$. 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.
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