Two-dimensional nonlinear optical response of a spiral magnet

We study the dynamical response function relevant for two-dimensional coherent nonlinear optical spectroscopy of the antiferromagnetic frustrated $J_{1}$-$J_{3}$ Heisenberg model on the square lattice within its long-range ordered, incommensurate diagonal spiral phase. We argue that in this phase effective dipole coupling to the electric field is important, with the spin-current coupling potentially being the dominant mechanism for spin-1/2. For this setting, we use linear spin wave theory to evaluate the leading nonlinear polarization response which is of second order in the driving field. We show that the response function features a strong antidiagonal, galvanoelectric feature. The width of this feature is set by relaxation rates beyond the noninteracting magnon picture, thereby providing access to single-magnon lifetimes within the multi-magnon continuum of the response function. Moreover, the response function is shown to display various structures in the two-dimensional frequency plane related to exceptional regions of the magnon dispersion.