For a <damped ferromagnetic spin wave>, shifting both poles into the lower half-plane preserves <causality>:
$$
G_R(\omega,k)=\frac{\omega_k/s}{(\omega+i\Gamma_k)^2-\omega_k^2}.
$$
Its <spectral function> is
$$
\varrho(\omega,k)=
\frac{4(\omega_k/s)\omega\Gamma_k}
{(\omega^2-\omega_k^2-\Gamma_k^2)^2+4\omega^2\Gamma_k^2}.
$$
The classical <Fluctuation-dissipation theorem> then gives
$$
S(\omega,k)=
\frac{4T(\omega_k/s)\Gamma_k}
{(\omega^2-\omega_k^2-\Gamma_k^2)^2+4\omega^2\Gamma_k^2}.
$$
Hence the zero-frequency <static structure factor> is
$$
S(k)=\frac{4T\omega_k\Gamma_k}
{s(\omega_k^2+\Gamma_k^2)^2},
$$
which has the stated proportionality after absorbing the normalization $4/s$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 337 / 2 / g / Solution

A damped ferromagnetic spin wave has a complex pole $\omega=\mathord\pm\omega_k-i\Gamma_k$. Both poles of its <retarded Green function> must lie in the lower half of the complex frequency plane so that the response respects <causality>.

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