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 a Transverse-field Ising model whose field term is $-2Jg{\sum }_n{S}_n^z$, the Hellmann–Feynman theorem gives the displayed relation. Differentiating $E_0=-J{\sum }_k\sqrt{1+g^2-2g\cos k}$ gives ${\sum }_n⟨S_n^z⟩={\sum }_k(g-\cos k)/(2\sqrt{1+g^2-2g\cos k})$. This is polarization along the imposed field, not spontaneous order along the exchange axis. In the thermodynamic limit it is zero per site at g=0, tends to1/2 at large positive g and equals $1/\pi$ at g=1.

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.

A bulk quasiparticle energy is not automatically the first excitation gap of a finite periodic spin chain. The Jordan–Wigner transformation fixes a boundary condition through fermion parity and restricts occupations to that same parity. At zero transverse field an Ising domain wall costs 2J, but periodic spin configurations contain an even number of walls; the first wall-pair excitation costs 4J. An ordered finite chain also has a distinct small symmetry-sector splitting at nonzero field. The thermodynamic dispersion and these finite-system scales answer different questions.