Critical transverse polarization of an Ising chain

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.