 Solution

For the even periodic chain, every classical Néel state ${\mathbf{S}}_m=(-1{)}^{m}S\mathbf{n}$ has energy $-JN{S}^2$, with arbitrary unit vector $\mathbf{n}$. Thus there is a continuous sphere of classical orientations, not merely two Ising-like states. Opposite staggered directions are connected by a global spin rotation.

A quantum Néel product is not an eigenstate: transverse exchange flips opposite neighbouring spins. The ground state therefore contains quantum superpositions and zero-point spin waves, reducing the classical staggered moment. A finite isotropic balanced chain has a symmetry-invariant ground state; a symmetry-broken reference is an approximation for analysing fluctuations, not its exact finite-system eigenstate.