Many-Body Vertex Effects: Time-Dependent Interaction Kernel with Correlated Multi-Excitons in the Bethe-Salpeter Equation

Building on a beyond-GW many-body framework that incorporates higher-order vertex effects in the self-energy -- giving rise to T-matrix and second-order exchange contributions -- this approach is extended to now include the vertex derived in that work to the kernel in the Bethe-Salpeter Equation (BSE) for the reducible polarization function. This results in a frequency-dependent interaction kernel that naturally captures random phase approximation (RPA) effects, dynamical excitonic interactions, and the correlated propagation of multiple correlated electron-hole pairs that model multi- (including bi- and tri-) excitonic effects, relevant for nonlinear optics and high harmonic generation. These processes emerge from including the functional derivatives of the screening and vertex with respect to the Green's function in the vertex, enabling a fully abinitio, time-dependent treatment of correlation effects. By focusing on the reducible rather than irreducible polarization function, this approach provides a computationally viable framework for capturing complex many-body interactions for calculating the self-energy, optical spectra and EELS. The resulting interaction kernel is relatively straightforward, clearly delineates the physical processes that are included and omitted, and has the same dimensionality as the conventional BSE kernel used in standard many-body theory implementations, but is now itself frequency dependent. The method is expected to facilitate the integration of advanced many-body effects into state-of-the-art software packages, offering a universal and highly accurate framework for the description of sub-atomic correlations. Such advancements are crucial for the development of semiconductor, optoelectronic, superconducting and antimatter technologies, and ensuring that theoretical modeling evolves alongside exascale and accelerated computing.