Many-body vertex effects: Time-dependent interaction kernel with correlated multiexcitons in the Bethe-Salpeter equation

Building on a beyond-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>G</a:mi><a:mi>W</a:mi></a:mrow></a:math> many-body perturbative framework that incorporates higher-order vertex effects in the self-energy—giving rise to <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>T</b:mi></b:math>-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 effects, dynamical excitonic interactions, and the correlated propagation of multiple correlated electron-hole pairs that model multiexcitonic (including bi- and triexcitonic) effects, relevant for nonlinear optics and high-harmonic generation. These processes emerge as a result of including the functional derivatives of the screening and vertex with respect to the Green's function in the vertex, enabling a fully , 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 electron energy loss spectroscopy. 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 perturbation 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 subatomic 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.