 Odd-momentum fermionic Bogoliubov angle

For a spinless Bogoliubov transformation $c_p=\cos {\theta }_p{\alpha }_p+i\sin {\theta }_p{\alpha }_{-p}^{†}$, consistency of the doubled Nambu spinor requires $\cos {\theta }_{-p}=\cos {\theta }_p$ and $\sin {\theta }_{-p}=-\sin {\theta }_p$. Thus choose ${\theta }_{-p}=-{\theta }_p$ modulo $2\pi$. The cross anticommutator from the first-row formulas is $\{c_p,c_{-p}\}=i\sin ({\theta }_p+{\theta }_{-p})$; a nontrivial even angle would violate the canonical anticommutation relations. A self-paired zero mode must instead have vanishing mixing in this convention. Real-angle unitarity within each two-by-two block alone does not enforce consistency between duplicated momentum labels.