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We clarify the pairing mechanism of high-$T_c$ superconductivity in bilayerLa$_3$Ni$_2$O$_7$ under high pressure by employing the static auxiliary fieldMonte Carlo approach to simulate a minimal effective model that contains local$d_{z^2}$ interlayer spin singlets and metallic $d_{x^2-y^2}$ bands.Superconductivity is induced when the local spin singlet pairs are mobilizedand attain long-distance phase coherence by hybridization with the metallicbands. When projected onto realistic Fermi surfaces, it yields a nodeless$s$-wave gap on the $\gamma$ Fermi surface, and extended $s$-wave gaps of thesame (opposite) sign on the $\alpha$ ($\beta$) Fermi surface due to its bonding(antibonding) character, with nodes or gap minima along the diagonal directionof the two-dimensional Brillouin zone. We find a dual role of the hybridizationthat not only induces global phase coherence but also competes with the spinsinglet formation. This lead to a tentative phase diagram where $T_c$ variesnonmonotonically with the hybridization, in good correspondence withexperimental observations. A roughly linear relation is obtained for realistichopping and hybridization parameters: $T_c\approx 0.04-0.05 J$, where $J$ isthe interlayer superexchange interaction. We emphasize the peculiar tunabilityof the bilayer structure and propose that $T_c$ may be further enhanced by holedoping or applying uniaxial pressure along the $c$ axis on superconductingLa$_3$Ni$_2$O$_7$. Our work provides reliable numerical evidence for thepairing mechanism of high-$T_c$ superconductivity in La$_3$Ni$_2$O$_7$ andpoints out a potential route to achieve even higher $T_c$.

High-$T_c$ superconductivity by mobilizing local spin singlets and possible route to higher $T_c$ in pressurized La$_3$Ni$_2$O$_7$