Many-Body effects beyond excitons in second-harmonic generation of monolayer MoS$_{2}$

We present a quantitative study of many-body effects including the three-particle level on second-harmonic generation in monolayer MoS$_2$. Our approach combines many-body perturbation theory with time-dependent current-density-functional theory within an \textit{ab initio} framework in the optical limit. The inclusion of two-particle excitonic effects \textit{via} a dynamical long-range linear exchange-correlation kernel reproduces the qualitative features of the second-harmonic response, but underestimates the experimentally reported magnitude by nearly a factor of two. By incorporating three-particle trionic correlations through a static long-range quadratic exchange-correlation kernel, we achieve significantly improved quantitative agreement with experiment. These findings highlight the role of many-body interactions beyond the excitonic level in accurately describing second-order optical responses in two-dimensional semiconductors.