Relaxation in Antiferromagnets due to Spin-Wave Interactions

For an antiferromagnet it is shown that within perturbation theory the Holstein‐Primakoff and Dyson‐Maleev transformations do not lead to identical results for either the static or dynamic properties. By examining the spin Green's functions we justify the use of the Dyson‐Maleev transformation when there are few spin waves present. Using second‐order perturbation theory we find the antiferromagnetic resonance linewidth to be $${\mathrm{\Delta \omega }}_0{\text{=(64ω}}_{{\mathrm{A\omega }}_0}{\mathrm{/\pi }}^3{S}^2{\omega }_E{\mathrm{)(kT/\hslash \omega }}_E{)}^2\exp {\mathrm{(-\hslash \omega }}_0\mathrm{/kT)\hspace{0.17em}\hspace{0.17em}}\mathrm{for}{\mathrm{\hspace{0.17em}\hspace{0.17em}kT\ll \hslash \omega }}_0$$ and $${\mathrm{\Delta \omega }}_0{\text{=[40ω}}_{\mathrm{A\zeta }}{\text{(3)/π}}^3{S}^2{\mathrm{](kT/\hslash \omega }}_E{)}^3\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{for}{\mathrm{\hspace{0.17em}\hspace{0.17em}\hslash \omega }}_0{\mathrm{\ll kT\ll \hslash \omega }}_E,$$ in qualitative agreement with the experimental results for MnF₂.