Superconductivity in the pressurized nickelate La$_3$Ni$_2$O$_7$ in the vicinity of a BEC-BCS crossover

Ever since the discovery of high-temperature superconductivity in cuprates, gaining microscopic insights into the nature of pairing in strongly correlated, repulsively interacting fermionic systems has remained one of the greatest challenges in modern condensed matter physics. Following recent experiments reporting superconductivity in the bilayer nickelate La$_3$Ni$_2$O$_7$ (LNO) with remarkably high critical temperatures of $T_c = 80$ K, it has been argued that the low-energy physics of LNO can be described by the strongly correlated, mixed-dimensional bilayer $t-J$ model. Here we investigate this bilayer system and utilize density matrix renormalization group techniques to establish a thorough understanding of the model and the magnetically induced pairing through comparison to the perturbative limit of dominating inter-layer spin couplings. In particular, this allows us to explain appearing finite-size effects, firmly establishing the existence of long-range superconducting order in the thermodynamic limit that is described by the XXZ universality class of hard-core bosons constituted by $s$-wave singlet pairs. As the effective model in the perturbative limit is known to show linear resistivity above the superconducting transition temperature, we propose a pair-based interpretation of the extended strange metal phase observed in LNO. By analyzing binding energies, we predict a BEC-BCS crossover as a function of the Hamiltonian parameters, whereas LNO is anticipated to lie on the BCS side in vicinity of the transition. We find large binding energies of the order of the inter-layer coupling that suggest strikingly high critical temperatures of the Berezinskii-Kosterlitz-Thouless transition, raising the question whether (mixD) bilayer superconductors possibly facilitate critical temperatures above room temperature.