Phase diagram of nickelate superconductors calculated by dynamical vertex approximation

We review the electronic structure of nickelate superconductors with and without effects of electronic correlations. As a minimal model we identify the one-band Hubbard model for the Ni 3$d_{x^2-y^2}$ orbital plus a pocket around the $A$-momentum. The latter however merely acts as a decoupled electron reservoir. This reservoir makes a careful translation from {nominal} Sr-doping to the doping of the one-band Hubbard model mandatory. Our dynamical mean-field theory calculations, in part already supported by experiment, indicate that the $\Gamma$ pocket, Nd 4$f$ orbitals, oxygen 2$p$ and {the} other Ni 3$d$ orbitals are not relevant in the superconducting doping regime. The physics is completely different if topotactic hydrogen is present or the oxygen reduction is incomplete. Then, a two-band physics hosted by the Ni 3$d_{x^2-y^2}$ and 3$d_{3z^2-r^2}$ orbitals emerges. Based on our minimal modeling we calculated the superconducting $T_c$ vs. Sr-doping $x$ phase diagram prior to experiment using the dynamical vertex approximation. For such a notoriously difficult to determine quantity as $T_c$, the agreement with experiment is astonishingly good. The prediction that $T_c$ is enhanced with pressure or compressive strain, has been confirmed experimentally as well. This supports that the one-band Hubbard model plus an electron reservoir is the appropriate minimal model.