Quantum depletion of Néel order ($M_s=S-N^{-1}{\sum }_k{v}_k^2$)

For a linear spin-wave approximation, the ground state is empty of the diagonal quasiparticles but contains the original spin bosons. If $a_k=u_k{\alpha }_k-v_k{\alpha }_{-k}^{†}$, their occupation number is $v_k^2$. The staggered magnetization is reduced from $S$ to $S-N^{-1}{\sum }_k{v}_k^2$. For the one-dimensional nearest-neighbor Heisenberg antiferromagnet, $v_k^2=(|\sin k{|}^{-1}-1)/2$ and the integral diverges logarithmically. This invalidates finite ordered magnetization within the approximation; it is not itself a determination of the exact spectral gap.

At zero temperature, linear-mode vacuum depletion scales as $\int dk{k}^{d-2}$; at nonzero temperature Bose occupation changes this to $\int dk{k}^{d-3}$. The two convergence criteria are different.

For linear spin waves, $\delta S=(2N{)}^{-1}{\sum }_k[(1-{\gamma }_k^2{)}^{-1/2}-1]$. The chain has ${\gamma }_k=\cos k$ and a logarithmic infrared divergence. This signals failure of the ordered expansion, not a physical negative staggered moment.