P3: Extending the GW approximation with the functional renormalization group
The basic idea of this project is that the well-known GW approximation (GWA), a standard workhorse [1,2] in the ab-initio description of spectra of solids, can very likely be improved by the use of the functional renormalization group (FRG) . In recent works, the group by S. Blügel set up a combination of GWA and the GT approximation (GTA) to sum up additional diagrams corresponding to magnetic fluctuations. This allowed them to study, from first principles, the properties of ferromagnetic spin wave excitations in transition metals  and spin-resolved quasiparticle self-energies showing clear signatures of the magnet-ic excitations. Nevertheless, despite the physical merits of GWA and GTA, such a treatment does not represent an unbiased approach that treats all fluctuation channels equally. In contrast with the GWA or GTA, in FRG  all one-loop fluctuations channels are summed up on equal footing. FRG is thus better suited in situations with an interplay of charge, magnetic, and pairing fluctuations. Moreover, both the GWA and the GTA can be identified as certain simplifications of the full diagrammatic content of the FRG. In the previous funding period of the RTG we have laid the foundations for further progress in this idea by the following steps:
1) The development of a highly parallelizable multiband FRG code that can be used in three spatial dimensions, i.e. the standard setting for ab-initio studies, and in which ab-initio model parameters including spin-orbit coupling  can be fed in smoothly,
2) the manifestations that in this scheme, the FRG adds important physics to the GWA results, and
3) the development and testing of simplified treatments of the frequency dependence of the effective interactions such that relevant self-energies can be computed and multiband prob-lems become tractable .
In the next steps we plan to assemble these elements from the first funding period into a scheme that can be used to compute observables that allow us to understand the differences obtained by using the FRG on the low energy scale in the band near the Fermi level instead of the GWA. A parameter-free low-energy Hamiltonian including effective interaction parameters [7,8] may constitute the input for these calculations. Beyond a proof of principle for the new method, we want to address material systems of interest. Specifically, we plan to study iron superconductors such as FeSe and other layered transition metal compounds that are known to exhibit an interplay of charge, nematic, and magnetic fluctuation channels. This physics is not transparently captured by established first-principles methods like GWA and GTA, which currently have to be augmented by other (often mean-field) approaches , but will be detectable by the FRG-enhanced first-principles approach.
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