Theoretical analysis of magnetic Raman scattering in ${\mathrm{La}}_2{\mathrm{CuO}}_4:$ Two-magnon intensity with the inclusion of ring exchange

We evaluate the Raman light-scattering intensity for the square lattice Heisenberg antiferromagnet with plaquette ring exchange $J_{\square }.$ With the exchange couplings as fixed before from an accurate fit to the spin-wave dispersion in ${\mathrm{La}}_2{\mathrm{CuO}}_4,$ leading in particular to $J_{\square }=0.24J,$ we demonstrate in a parameter-free calculation that the inclusion of the plaquette exchange contribution to the dispersion and the magnon-magnon interaction vertex gives a peak position in $B_{1g}$ scattering geometry $E_{\max }=2.71J$ which is in excellent agreement with the experimental data. Yet, the intrinsic width and the line shape of the two-magnon remain beyond a description in terms of a spin-only Hamiltonian.