Correlated electronic structure of La$_3$Ni$_2$O$_7$ under pressure

Recently, superconductivity with a $T_c$ up to 78 K has been reported in bulk samples of the bilayer nickelate La$_3$Ni$_2$O$_7$ at pressures above 14 GPa. Important theoretical tasks are the formulation of relevant low-energy models and the clarification of the normal state properties. Here, we study the correlated electronic structure of the high-pressure phase in a four-orbital low-energy subspace using different many-body approaches: $GW$, dynamical mean field theory (DMFT), extended DMFT (EDMFT) and $GW$+EDMFT, with realistic frequency-dependent interaction parameters. The nonlocal correlation and screening effects captured by $GW$+EDMFT result in an instability towards the formation of charge stripes, with the $3d_{z^2}$ as the main active orbital. We also comment on the potential relevance of the rare-earth self-doping pocket, since hole doping suppresses the ordering tendency.