Ising-chain field derivative and transverse polarization (${\sum }_n⟨S_n^z⟩=-{\partial }_g{E}_0/(2J)$)

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 the convention used in Ising-chain field derivative and transverse polarization, the polarization per site at the critical field $g=1$ isIndeed, $(g-\cos k)/(2\sqrt{1+g^2-2g\cos k})$ becomes $|\sin (k/2)|/2$, whose Brillouin-zone average is $1/\pi$. The full-polarization value in this spin normalization is $1/2$, so criticality does not mean complete alignment with the transverse field.

This expectation value is the response along the imposed field obtained by differentiating the ground-state energy. It is distinct from the spontaneous order along the exchange axis. Confusing those two directions would give the wrong interpretation of a finite transverse polarization at the quantum critical point.