Correlation effects in solids: From DFT to DMFT

In this lecture we give an introduction to theoretical description of strongly correlated materials based on dynamical mean-field theory (DMFT) and its extensions. The goal of this theoretical construction is to retain the many-body aspects of local atomic physics within the extended solid. The effects of short-range non-local correlations within cluster extensions of the DMFT scheme, as well as long-range fluctuations within the fully renormalized dual-fermion perturbation scheme are discussed extensively. Recent progress in the numerical solution of the DMFT effective quantum impurity problem within the recently developed continuous-time Quantum Monte Carlo schemes is reviewed. We then describe realistic extensions of this approach which combine the accuracy of first-principle Density-Functional Theory with the treatment of local many-body effects within DMFT.