 Solution

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.