The large interaction between electrons that is seen in some materials leads to a variety of intriguing and still poorly understood phenomena, like e.g. Mott insulators or high-temperature superconductivity. These strongly-interacting systems raise a vast spectrum of questions, from fundamental issues (e.g. what is the mechanism giving rise to high-temperature superconductivity?) to material-science issues (e.g. how to modify a material to increase its superconducting temperature?). From the theoretical point of view, understanding these materials requires to describe the collective behaviour of a large number of interacting electrons (the many-body problem). While progress has been made, one of the main challenges in the field is to construct predictive and controlled theoretical approaches, suited for both simple models and realistic material computations.
The goal of this thesis is to work on developing a new generation of theoretical approaches and of innovative algorithms for the many-body problem and to apply them to simple models for high-temperature superconductors. The thesis will thus contain three aspects:
i) construction and study of new formalisms beyond dynamical mean-field theory methods;
ii) elaboration and implementation of new algorithms using modern programming techniques (within the TRIQS project), including some high-performance computing;
iii) application of these new tools to describe the phenomenology of strongly-correlated materials and comparison with experiments.