Beyond the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>G</mml:mi><mml:mi>W</mml:mi></mml:mrow></mml:math>approximation: Combining correlation channels

In many-body perturbation theory (MBPT) the self-energy $\ensuremath{\Sigma}=iGW\ensuremath{\Gamma}$ plays a key role since it contains all the many-body effects of the system. The exact self-energy is not known; as a first approximation one can set the vertex function $\ensuremath{\Gamma}$ to unity which leads to the GW approximation. The latter properly describes the high-density regime, where screening is important; in the low-density regime, instead, other approximations are proposed, such as the $T$ matrix, which describes multiple scattering between two particles. Here we combine the two approaches. Starting from the fundamental equations of MBPT, we show how one can derive the $T$-matrix approximation to the self-energy in a common framework with $GW$. This allows us to elucidate several aspects of this formulation, including the origin of, and link between, the electron-hole and the particle-particle $T$ matrix, the derivation of a screened $T$ matrix, and the conversion of the $T$ matrix into a vertex correction. The exactly solvable Hubbard molecule is used for illustration.