Self-consistency and vertex corrections beyond the GW approximation

The good performance of the GW approximation for band-structure calculations in solids was long taken as a sign that the sum of self-energy diagrams is converged and that all omitted terms are small. However, with modern computational resources it has now become possible to evaluate self-consistency and vertex corrections explicitly, and the numerical results show that they are, in general, not individually negligible. In this review the available data is examined, and the implications for practical calculations and the theoretical foundation of the GW approximation are discussed.