Ab initio methods for superconductivity

Modern ab initio theories of superconductivity allow characterizing and predicting phonon-mediated superconductors. In this Technical Review, we analyse Eliashberg theory, density functional theory for superconductors as well as McMillan and Allen-Dynes equations, providing a summary of the underlying approximations and capabilities. We highlight in simple terms and with examples the many sources of error, which may lead to inaccurate predictions, including limitations on the applicability of the methods, subtle convergence aspects and improper practices often adopted to simplify the treatment of Coulomb interactions. Additionally, we compare the accuracy of the various methods by computing the critical temperature (Tc) for a broad range of superconductors and benchmarking against experimental results. We find that even the simple McMillan and Allen-Dynes formulas give Tc distributions centred on the experimental values. The Eliashberg theory and density functional theory for superconductors yield more peaked distributions, strongly reducing the possibility of incorrect predictions. Ab initio theories of superconductivity allow characterizing and predicting phonon-mediated superconductors. This Technical Review provides an analysis of the different theories, highlighting the main sources of error, either due to inherent approximations or arising from improper practices, and provides a comparison against experimental results.