Merging of dynamical mean-field theory and functional renormalization group

The theoretical description of correlated electron systems beyond the perturbative regime represents one of the main challenges for the forefront research in condensed matter physics. In fact, the impressively fast progress in the experimental engineering of correlated electron properties from bulk systems down to the nanoscale is not fully balanced yet by corresponding advances in the theoretical tools to describe them. This project aims at filling this gap with the algorithmic implementation of a novel scheme recently proposed by the applicants in [Phys. Rev. Lett. 112, 196402 (2014)], consisting in the combination of two of the most successful quantum many-body methods: the dynamical mean field theory (DMFT) and the functional renormalization group (fRG). In spite of their success, in fact, both methods have significant limitations: due to its mean-field description, DMFT neglects all non- local spatial correlations, while it captures the local part of the electronic correlations which drive, e.g., the Mott metal-insulator transitions. In contrast, the non-local correlations can be efficiently tackled by the fRG, whose application is, however, typically limited to the pertubative regime of weak electronic correlations. Our novel approach, coined DMF2RG, aims at overcoming the restrictions of both, by using the DMFT solution as starting point of the fRG treatment. This way, local - and possibly strong - correlations are fully taken into account from the very beginning within DMFT, while non-local correlations will be systematically included through the fRG procedure. By exploiting the complementary strengths of the existing state-of-the-art approaches, the DMF2RG represents a breakthrough for the theory of correlated electrons and its applications. The proposed project includes an efficient algorithmic implementation of the DMF2RG idea, its benchmarking against other methods and limiting cases and, finally, its application to prototype models and realistic systems. In particular, we aim at providing a new insight on the competing physical mechanisms at work in systems of high scientific interests, such as the pseudogap phases of lightly doped Mott insulators, unconventional superconductors and systems of adatoms on semiconducting surfaces. A long-term perspective is the multi-orbital implementation of the DMF2RG in view of a broad application of our new method by the whole solid state physics community.

The main aim of our project was the algorithmic implementation of an approach merging two of the most powerful methods (the dynamical mean-field theory and the functional renormalization group) for treating correlation effects among electrons in quantum materials in order to allow for reliable numerical many-body investigations in their most challenging (and, often, physically most interesting) parameter regions. Calculations by means of this combined approach have been indeed performed in all coupling regimes of fundamental many-electron models, such as the two-dimensional Hubbard model. While the exciting results obtained have outlined how to extend our scheme in a well-defined direction, the intensive effort made to improve the computational procedure, whose details have been presented in several publications, has allowed significant advancements also beyond the original scope of the project. From an algorithmic perspective, the numerical improvements we have introduced to treat more efficiently the building block of our approach, i.e., the electronic scattering amplitude, turned out to be of high interest to the scientific community working on the theoretical treatment of many-electron physics, as demonstrated by their inclusions in advanced algorithms of other research groups. To be also mentioned, in this respect, is the recent ``proof of principle'' of how machine learning tools can be applied to our class of algorithms, whose first results appear very promising for future method development. At the same time, unexpected substantial progress has also been obtained on a more fundamental physical level. In fact, by learning how to read relevant physical properties directly from the main frequency-structures of the electronic scattering functions mentioned above, we have been able to unveil, for the first time, a direct link between important formal aspects of the many-electron theory in the strong-coupling regime and the underlying physics: We could ascribe the physical origin of the breakdown of textbook many-electron methods, such as the conventional self-consistent perturbation expansion, to the formation of local magnetic moments and, specifically, to the way this affects the local fluctuations of the electronic density. This new insight has also allowed us to identify a new criterion for estimating the Kondo-temperature at which magnetic moments get screened in correlated electron systems, uniquely based on the low-energy dynamical properties of the local fluctuations of the electronic charge. Moreover, we discovered how electronic processes, which cannot be described in a conventional perturbative framework, can eventually result in a surprising sign-flip of the effective coupling (from repulsive to attractive) between the electrons, responsible for the phase-separation instabilities occurring in the proximity of metal-insulator-transitions driven by electronic correlation. This kind of non-perturbative microscopical processes might also play a pivotal role for the onset of unconventional high-temperature superconductivity in correlated quantum materials, certainly deserving focused future investigations.