 Canonical rotation of two fermionic modes

A real two-mode Bogoliubov transformation preserving the full canonical anticommutation relations. The self anticommutators become $\{a,a^{†}\}=\{b,b^{†}\}=u^2+v^2$, while $\{a,b\}=-uv+uv=0$ and every other cross anticommutator vanishes. Thus its matrix is an orthogonal matrix. For the two-orbital fermionic pairing Hamiltonian, let $E=\sqrt{{\epsilon }^2+{\Delta }^2}$, take $u=\sqrt{(E+\epsilon )/(2E)}$ and $v=\sqrt{(E-\epsilon )/(2E)}$, and obtain $H=E(c^{†}c+d^{†}d)+\epsilon -E$. Inverting gives $c=ua-v{b}^{†}$, $d=v{a}^{†}+ub$.