The Eliashberg equation for the anomalous self-energy of the superconducting state is a powerful tool for the study of superconductivity by both conventional (electron-phonon) and unconventional (purely electronic) mechanisms. In the latter case, however, Migdal's theorem on the unimportance of vertex correction fails and care must be exerted in constructing the effective electron-electron interaction. In this talk I review theoretical studies of purely electronic superconductivity which I have been involved with over the years in systems as diverse as the electron hole liquid [1], the two-dimensional electron gas [2], and twisted bilayer graphene [3]. Vertex corrections (included approximately in the form of local field factors) drive superconductivity in the first case, suppress it in the second, and have not yet been evaluated in the third.

[1] G. Vignale and K. S. Singwi, Possibility of Superconductivity in the Electron-Hole Liquid, Phys. Rev. B 31, 2729 (1985).
[2] G. S. Canright and G. Vignale, Superconductivity and acoustic plasmons in the two-dimensional electron gas, Phys. Rev. B 39, 2740 (1989).
[3] G. Sharma, M. Trushin, O. P. Sushkov, G. Vignale, and S. Adam, Superconductivity from collective excitations in magic-angle twisted bilayer graphene, Phys. Rev. Research 2, 022040(R) (2020).