Ising-chain ground-state energy ($E_0=-\frac{1}{2}{\sum }_k{ϵ}_k$)

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.