= 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 $g=1$ is
$$
\frac{\langle S^z_{\rm tot}\rangle}{N}=\frac1\pi.
$$
Indeed, $(g-\cos k)/(2\sqrt{1+g^2-2g\cos k})$ 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>.
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