Motivated by cuprate and nickelate superconductors, we perform a comprehensive study of the superconducting instability in the single-band Hubbard model. We calculate the spectrum and superconducting transition temperature $T_c$ as a function of filling and Coulomb interaction for a range of hopping parameters, using the dynamical vertex approximation. We find the sweet spot for high $T_c$ to be at intermediate coupling, moderate Fermi surface warping, and low hole doping. Combining these results with first principles calculations, neither nickelates nor cuprates are close to this optimum within the single-band description. Instead, we identify some palladates, notably ${\mathrm{RbSr}}_2{\mathrm{PdO}}_3$ and $A_2^{\prime }{\mathrm{PdO}}_2{\mathrm{Cl}}_2$ ($A^{\prime }={\mathrm{Ba}}_{0.5}{\mathrm{La}}_{0.5}$), to be virtually optimal, while others, such as ${\mathrm{NdPdO}}_2$, are too weakly correlated.

Motivated by the recent discovery of superconductivity in the pentalayer nickelate ${\mathrm{Nd}}_6{\mathrm{Ni}}_5{\mathrm{O}}_{12}$ [G. A. Pan et al., Nat. Mater. 21, 160 (2022)], we calculate its electronic structure and superconducting critical temperature. We find that electronic correlations are essential for pushing ${\mathrm{Nd}}_6{\mathrm{Ni}}_5{\mathrm{O}}_{12}$ into the superconducting doping range as they shift the electron pockets above the Fermi energy. As a consequence, ${\mathrm{Nd}}_6{\mathrm{Ni}}_5{\mathrm{O}}_{12}$ can be described with a single $d_{x^2-y^2}$ orbital per Ni. We predict the pentalayer to be overdoped; further improvements in $T_c$ can be expected for a few additional layers. Instead, for the bilayer nickelate ${\mathrm{Nd}}_3{\mathrm{Ni}}_2{\mathrm{O}}_6$ we find correlations to drive the system into a three-orbital regime also involving the Ni $d_{xz,yz}$ states. We suggest, however, that single-orbital physics with optimal doping can be restored by substituting 60% of the trivalent Nd or La by tetravalent Zr.

Infinite-layer nickelates show high-temperature superconductivity, and the experimental phase diagram agrees well with the one simulated within the dynamical vertex approximation ($\mathrm{D}\mathrm{\Gamma }\mathrm{A}$). Here, we compare the spin-fluctuation spectrum behind these calculations to resonant inelastic x-ray scattering experiments. The overall agreement is good. This independent cross validation of the strength of spin fluctuations strongly supports the scenario, advanced by $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$, that spin fluctuations are the mediator of the superconductivity observed in nickelates.

We present an analytically solvable model for correlated electrons, which is able to capture the major Fermi surface modifications occurring in both hole- and electron-doped cuprates as a function of doping. The proposed Hamiltonian qualitatively reproduces the results of numerically demanding many-body calculations, here obtained using the dynamical vertex approximation. Our analytical theory provides a transparent description of a precise mechanism, capable of driving the formation of disconnected segments along the Fermi surface (the highly debated “Fermi arcs”), as well as the opening of a pseudogap in hole and electron doping. This occurs through a specific mechanism: The electronic states on the Fermi arcs remain intact, while the Fermi surface part where the gap opens transforms into a Luttinger arc.

Despite enormous experimental and theoretical efforts, obtaining generally accepted conclusions regarding the intrinsic magnetic and electronic properties of superconducting nickelates remains exceptionally challenging. Experiments show a significant degree of uncertainty, indicating hidden factors in the synthesized films, which call for further investigations. One of those hidden factors is the possibility of intercalating hydrogen during the chemical reduction process from $\mathrm{Nd}\left(\mathrm{La}\right){\mathrm{NiO}}_3$ to $\mathrm{Nd}\left(\mathrm{La}\right){\mathrm{NiO}}_2$ using ${\mathrm{CaH}}_2$. While hydrogen has been detected in experimental samples, not much is known about its distribution through the crystal and its influence on the electronic environment. Here, we show the tendency toward the formation of one-dimensional hydrogen chains in infinite-layers ${\mathrm{LaNiO}}_2$ superconductors using density-functional theory supplemented by dynamical mean-field theory. The formation of such hydrogen chains induces a coexistence of different oxidation states of Ni and competing magnetic phases, and possibly explains the recently observed charge order states in nickelate superconductors. Furthermore, it contributes to the difficulty of synthesizing homogeneous nickelates and determining their ground states. The smoking gun to detect excess hydrogen in nickelates is the flat phonon modes, which are infrared active and quite insensitive to the exact arrangement of the H atoms.

Superconductivity has entered the nickel age marked by enormous experimental and theoretical efforts. Notwithstanding, synthesizing nickelate superconductors remains extremely challenging, not least due to incomplete oxygen reduction and topotactic hydrogen. Here, we present density-functional theory calculations for nickelate superconductors with additional topotactic hydrogen or oxygen, namely La${}_{1-x}$Sr${}_x$NiO${}_2$H${}_{\delta }$ and LaNiO${}_{2+\delta }$. We identify a phonon mode as a possible indication for topotactic hydrogen and discuss the charge redistribution patterns around oxygen and hydrogen impurities.