Logarithmic quantum depletion in a Heisenberg chain
= 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.