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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.

The Hellmann–Feynman theorem applied to the source field normalization givesDifferentiating the Ising-chain ground-state energy, using ${\partial }_g{ϵ}_k=4J^2(g-\cos k)/{ϵ}_k$, therefore provesThis is the Ising-chain field derivative and transverse polarization, not its spontaneous x-order parameter. It tends to zero at g=0 and to N/2 at large positive field. The critical transverse polarization of an Ising chain is finite: in the thermodynamic critical limit this gives $⟨S^z⟩=1/\pi$. A zero-energy isolated momentum is treated by the appropriate limiting state, rather than by assigning a value to a literal 0/0 summand.