Ising-chain Bogoliubov diagonalization (${ϵ}_k=2J\sqrt{1+g^2-2g\cos k}$)

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 $H={\sum }_k{ϵ}_k({\gamma }_k^{†}{\gamma }_k-1/2)$. 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.

The vacuum of the diagonal Bogoliubov quasiparticles has the displayed energy. In the thermodynamic bulk, $E_0/N=-J(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }\sqrt{1+g^2-2g\cos k}dk$. The Hellmann–Feynman theorem then gives the field magnetization ${\sum }_n⟨S_n^z⟩={\sum }_{k}J(g-\cos k)/{ϵ}_k$. Sector-dependent finite-chain boundary terms must be restored when exact finite-size levels are needed.

For nonnegative field $g$, the bulk quasiparticle gap is $2J|g-1|$. At $g=0$ the dispersion is flat, $2J$; at large $g$ it is $2Jg-2J\cos k+O(J/g)$. At $g=1$ it becomes $4J|\sin (k/2)|$, linear near the gapless point. The finite periodic spin chain has additional global excitation constraints; a single isolated domain wall is not an allowed closed-chain spin configuration.