Momentum-dependent light scattering in insulating cuprates

We investigate the problem of inelastic x-ray scattering in the spin-$1∕2$ Heisenberg model on the square lattice. We first derive a momentum-dependent scattering operator for the $A_{1g}$ and $B_{1g}$ polarization geometries. On the basis of a spin-wave analysis, including magnon-magnon interactions and exact diagonalizations, we determine the qualitative shape of the spectra. We argue that our results may be relevant to help interpret inelastic x-ray scattering experiments in the antiferromagnetic phase of the cuprates.

Figure 1

(Color online) Spin-wave intensity spectrum without [$I_0(\mathbf{q},\omega )$, left] and with [$I_{\mathrm{RPA}}(\mathbf{q},\omega )$, right] magnon-magnon interaction. The color intensity scale is 2.2 times smaller for the right panel than the left.

Figure 2

(Color online) 16-site cluster: exact diagonalizations for $A_{1g}$ and $B_{1g}$ polarizations, to be compared with spin-wave results. The spectra are shown for the independent momenta $\mathbf{q}$ allowed for the $N=16$-site cluster (given on the side of the rightmost panel). The relative intensity of the curves is given by $⟨\mathcal{O}(\mathbf{q}{)}^2⟩$.

Figure 3

(Color online) 20-site cluster: exact diagonalizations in the $B_{1g}$ channel. The inset is a schematic of the investigated $\mathbf{q}$ points.