= Solution
The inverse transformation has negative off-diagonal entries. Choosing $\tanh2\theta_k=\gamma_k$ removes anomalous terms and gives
$$
\boxed{\omega_k=2JS\sqrt{1-\gamma_k^2}=2JS|\sin k|,\qquad H=-JNS(S+1)+\sum_k\omega_k(\alpha_k^\dagger\alpha_k+1/2)+O(S^0).}
$$
There are gapless points at $k=0$ and the zone boundary; near zero, $\omega_k\simeq2JS|k|$, a linear <antiferromagnetic spin wave> with velocity $2JS$ in the stated units. The zero modes are collective rotations: their divergent squeezing cannot be treated as ordinary normalized independent oscillator vacua. A finite-size cutoff or infinitesimal symmetry-breaking field regulates them.
As a constant-term check, the continuum spin-wave vacuum energy per site is $-JS(S+1)+(2/\pi)JS$. This is a spin-wave prediction, not an exact spectrum for every finite-$S$ one-dimensional chain.
\Image[/past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2014/paper-1-qcm2014-spin-waves.png]
{title=Linear antiferromagnetic spin-wave dispersion and logarithmically growing one-dimensional vacuum depletion}
{source=paper-1-qcm2014-spin-waves.py}
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