= Ising-chain field derivative and transverse polarization
{c}
{title2=$\sum_n\langle S_n^z\rangle=-\partial_gE_0/(2J)$}
For a <Transverse-field Ising model> whose field term is $-2Jg\sum_nS_n^z$, the <Hellmann–Feynman theorem> gives the displayed relation. Differentiating $E_0=-J\sum_k\sqrt{1+g^2-2g\cos k}$ gives $\sum_n\langle S_n^z\rangle=\sum_k(g-\cos k)/(2\sqrt{1+g^2-2g\cos k})$. 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.
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