Exploring new exchange-correlation kernels in the Bethe-Salpeter equation: a study of the asymmetric Hubbard dimer

The Bethe-Salpeter equation (BSE) is the key equation in many-body perturbation theory based on Green's functions to access response properties. Within the $GW$ approximation to the exchange-correlation kernel, the BSE has been successfully applied to several finite and infinite systems. However, it also shows some failures, such as underestimated triplet excitation energies, lack of double excitations, ground-state energy instabilities in the dissociation limit, etc. In this work, we study the performance of the BSE within the $GW$ approximation as well as the $T$-matrix approximation for the excitation energies of the exactly solvable asymmetric Hubbard dimer. This model allows one to study various correlation regimes by varying the on-site Coulomb interaction $U$ as well as the degree of the asymmetry of the system by varying the difference of potential $\Delta v$ between the two sites. We show that, overall, the $GW$ approximation gives more accurate excitation energies than $GT$ over a wide range of $U$ and $\Delta v$. However, the strongly-correlated (i.e., large $U$) regime still remains a challenge.