Abstract
Recent experiments on ${\mathrm{La}}_{2}\mathrm{Cu}{\mathrm{O}}_{4}$ suggest that indirect resonant inelastic x-ray scattering (RIXS) might provide a probe for transversal spin dynamics. We present in detail a systematic expansion of the relevant magnetic RIXS cross section by using the ultrashort core-hole lifetime (UCL) approximation. We compute the scattering intensity and its momentum dependence in leading order of the UCL expansion. The scattering is due to two-magnon processes and is calculated within a linear spin-wave expansion of the Heisenberg spin model for this compound, including longer range and cyclic spin interactions. We observe that the latter terms in the Hamiltonian enhance the first moment of the spectrum if they strengthen the antiferromagnetic ordering. The theoretical spectra agree very well with experimental data, including the observation that scattering intensity vanishes for the transferred momenta $\mathbf{q}=(0,0)$ and $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$. We show that at finite temperature, there is an additional single-magnon contribution to the scattering with a spectral weight proportional to ${T}^{3}$. We also compute the leading corrections to the UCL approximation and find them to be small, setting the UCL results on a solid basis. All this univocally points to the conclusion that the observed low temperature RIXS intensity in ${\mathrm{La}}_{2}\mathrm{Cu}{\mathrm{O}}_{4}$ is due to two-magnon scattering.
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