Theory of Raman scattering in layered cuprate materials

The line shape of Raman scattering in a two-dimensional Heisenberg antiferromagnet at zero temperature is studied within spin-wave theory, by using the Dyson-Maleev transformation. Besides the well-known dominant contribution coming from two-magnon scattering, a systematic analysis of the contribution coming from the production of four magnons is carried out. It is found that the four-magnon contribution is very small compared with the two-magnon intensity. However, it occurs at such large energies that the ratio of the first two frequency cumulants is enhanced by a factor of 2.5 for S=1/2, in fairly good agreement with recent series-expansion estimates. This is true despite the fact that the total line shape that we obtain is still in rather poor agreement with the experimental line shape obtained in experiments on undoped ${\mathrm{La}}_2$ ${\mathrm{CuO}}_4$. As a by-product of the analysis of the effects of spin-wave interactions on the magnon propagator, the renormalization factor of the spin-wave velocity is computed to O((1/S${)}^2$). For S=1/2, the number that we find is ${\mathit{Z}}_{\mathit{c}}$=1.177, in very good agreement with the series-expansion result.