= Solution
The vacuum of the diagonal <quasiparticles> contains original spin bosons: $\langle a_k^\dagger a_k\rangle=v_k^2$. Thus the <quantum depletion of Néel order> gives
$$
\boxed{M_s=S-\frac1{2N}\sum_k\left(\frac1{\sqrt{1-\gamma_k^2}}-1\right).}
$$
In one dimension $\sqrt{1-\gamma_k^2}=|\sin k|$. Removing the global zero modes with an infrared cutoff, the continuum integral contains $\int dk/|k|$ near both gapless points and diverges logarithmically as the cutoff is removed. This is a breakdown of the assumed <Néel order> reference, not a physical infinitely negative <magnetization>. It indicates that the one-dimensional ordered spin-wave expansion cannot maintain a finite staggered moment; it does not determine the exact <spectral gap> or all properties of the spin chain for arbitrary $S$.
At zero temperature in $d$ dimensions the singular contribution scales as $\int_0 k^{d-2}\,dk$. It is infrared finite for $d\ge2$, permitting a finite quantum reduction and a self-consistent ordered spin-wave description in an appropriate regime. This differs from the positive-temperature contribution, where $n_B(\omega)\sim T/\omega$ yields $\int_0 k^{d-3}\,dk$. Its divergence for $d\le2$ agrees with the <Mermin-Wagner theorem> for short-range continuous-symmetry models. Finite-temperature nonordering and the zero-temperature one-dimensional depletion argument are separate statements.
Back to article page
