= Canonical rotation of two fermionic modes
{title2=$a=uc+vd^\dagger,\quad b=-vc^\dagger+ud,\quad u^2+v^2=1$}
A real two-mode <Bogoliubov transformation> preserving the full <canonical anticommutation relations>. The self <anticommutators> become $\{a,a^\dagger\}=\{b,b^\dagger\}=u^2+v^2$, 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 $E=\sqrt{\varepsilon^2+\Delta^2}$, take $u=\sqrt{(E+\varepsilon)/(2E)}$ and $v=\sqrt{(E-\varepsilon)/(2E)}$, and obtain $H=E(c^\dagger c+d^\dagger d)+\varepsilon-E$. Inverting gives $c=ua-vb^\dagger$, $d=va^\dagger+ub$.
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