Dynamical Effective Hamiltonian Approach to Second-Harmonic Generation in Quantum Magnets: Application to NiI$_2$

Although second harmonic generation (SHG) is a promising and widely used method recently for studying 2D magnetic materials, the quantitative analysis of the full SHG tensor is currently challenging. In this letter, we describe a first-principles-based approach towards quantitative analysis of SHG in insulating magnets through formulation in terms of dynamical effective operators. These operators are computed by solving local many-body cluster models. We benchmark this method on NiI$_2$, a multiferroic 2D van der Waals antiferromagnet, demonstrating quantitative analysis of reported Rotational Anisotropy (RA)-SHG data. SHG is demonstrated to probe local ring-current susceptibilities, which provide sensitivity to short-range chiral spin-spin correlations. The described methods may be easily extended to other non-linear optical responses and materials.