Logarithmic quantum depletion in a Heisenberg chain

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.