= Ising-chain Bogoliubov diagonalization
{c}
{title2=$\epsilon_k=2J\sqrt{1+g^2-2g\cos k}$}
For a paired non-self-conjugate <momentum>, rotate the <Ising-chain Nambu spinor> by an angle with $\cos\theta=(g-\cos k)/r_k$, $\sin\theta=-\sin k/r_k$ and $r_k=\sqrt{1+g^2-2g\cos k}$. Opposite angles at $k,-k$ preserve the <canonical anticommutation relations>. The result is $H=\sum_k\epsilon_k(\gamma_k^\dagger\gamma_k-1/2)$. Self-paired <momenta> have no pairing and are handled directly as number levels. Periodic spin-chain sectors additionally constrain <fermion parity> and allowed mode quantization.
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