Amplitude-Renormalization Factor in Two-Magnon Raman Scattering

The Dyson-Maleev graphical approach to two-magnon Raman scattering in simple antiferro-magnets, which was developed in two previous papers, is extended to include the simplest class of corrections which arise from using the complete Raman-transition operator $M$ in place of the lowest-order approximation for $M$ considered previously. The class of diagrams considered yields an amplitude renormalization factor for the two-magnon Raman process. Previously Sólyom, using a different graphical method, has obtained an amplitude renormalization factor of ${〈S_a^z〉}^2$ where $〈S_a^z〉$ is the sublattice magnetization. Sólyom's result predicts a vanishing Raman intensity at the ordering temperature $T_N$. In the present paper it is shown that Sólyom's result is a consequence of including only the effects of the transverse ($S^x,S^y$) interactions in $M$, and that when the additional longitudinal ($S^z$) interactions in $M$ are included, the correct amplitude renormalization factor is ${\alpha }^2\mathrm{}\left(T\right)\mathrm{}$, where $\alpha \mathrm{}\left(T\right)\mathrm{}$ is the energy renormalization factor of the Hartree-Fock theory. Since $\alpha \mathrm{}\left(T\right)\mathrm{}$ remains finite at the ordering temperature, the unsatisfactory vanishing of the Raman amplitude at $T_N$ is eliminated. Although the amplitude-renormalization processes considered here do not improve the line-shape difficulties discussed previously, the fact that they are non-negligible suggests that more complicated processes which arise from using the complete expression for $M$ will also have a significant effect on both the amplitude and line shape of the higher-temperature spectra.