# P3: FRG extension of the GW approximation

## Project Description

Hedin's closed set of equations lays the foundation for the ab-initio investigation of electronic self-energies in materials. The lowest-order approximation to the full set of equations yields the *GW* approximation^{(1), (2)}. In particular, for semiconductors, this scheme has proved very successful for the description of single-particle excitations. Our implementation of the *GW* approximation in the all-electron software package SPEX ^{(3), (4)} opens the door to the application to more complex electronic materials as well, such as oxides^{ (5)}. In conjunction with the Wannier representation and down-folding, this allows for the quantitative calculation of Hubbard-*U* interaction parameters^{ (6)}. A central limitation of the *GW* approximations is, however, that the screening of the interactions is primarily caused by charge fluctuations, while, e.g., magnetic fluctuations are not resummed. There are first attempts to capture these as well, e.g. in the so-called* GT* approximation^{ (7), (8)}. In this project we follow a new route in this direction, based on the functional renormalization group, short FRG ^{(9)}. A strength of the FRG is that is resumes different classes of diagrams (among those screening, vertex corrections, and magnetic fluctuations) on the same footing, similar to parquet approximations ^{(10),(11)}. In its full form, the interaction then depends on three momentum and frequency variables, leading to a substantial numerical effort. Here, new developments within the FRG context ^{(12)-(15)} may provide some progress. In these recent works, the possibly strong dependence on the three variables is reduced to a strong dependence on only one variable, by the help of a form factor expansion and the use of three different channels to parameterize the interaction. This reduces the numerical effort considerably. First works in the two-dimensional Hubbard model and related multiband systems ^{(16)} show a good agreement with previous full-fledged treatments^{(17)- (21)}. Very importantly, with these clever approximations, also the inclusion of non-trivial self-energy effects is in each^{(22)}. Connecting these FRG schemes with the *GW* and the *GT* approximation becomes possible by noticing that the *GW* or *GT* schemes can be reproduced by constraining the FRG to a specific one out of the three channels mentioned above.

The goal of the present project is to carry over this FRG methodology to the framework of the *GW* ab-initio calculations. By this, a unified treatment and extension of the *GW* and *GT* approximations can be reached. The first step of the project will be to implement and test this idea in a simple system with one or two bands in a Wannier basis. Subsequently, we aim to integrate the FRG/*GW* extension into the ab-initio package SPEX, and therefore open the way for an application to more realistic material systems. This project will have a constructive overlap with project P1.

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(4) SPEX (http://www.judft.de/pm/index.php?n=Codes.Spex)

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*Erratum*: Phys. Rev. B **84**, 039906 (2011)

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Phys. Rev. B **85**, 035414 (2012);

W.-S. Wang, Z. Z. Li, Y. Y. Xiang, Q. H. Wang, Phys. Rev. B **87**, 115135 (2013)

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(22) K.-U. Giering and M. Salmhofer, arXiv:1208.6131 (2012)