{\it Ab initio} prediction of $d_{x^2-y^2}$-wave superconductivity in infinite-layer nickelates

Infinite-layer nickelates have recently emerged as a new family of potential unconventional high critical temperature ($T_c$) superconductors. However, fundamental questions such as their superconducting (SC) pairing mechanism and gap symmetry remain under intense debate. Here we present a fully {\it ab initio} theoretical study on the SC properties of optimally doped nickelates $Re$$_{0.8}$Sr$_{0.2}$NiO$_2$ ($Re=$ La, Pr, Nd), based on the density functional theory for superconductors calculations with electron-phonon coupling (EPC), screened Coulomb repulsion and spin fluctuation (SF) interaction treated on an equal footing. We find that $Re_{0.8}$Sr$_{0.2}$NiO$_2$ are two-band superconductors with sign reversal $d_{x^2-y^2}(\pm)$-wave gap functions on the different Fermi surface (FS) pockets. Interestingly, when the SF interaction is turned off, $T_c$ becomes negligibly small ($\sim$0.01 K), thus demonstrating that the superconductivity in $Re_{0.8}$Sr$_{0.2}$NiO$_2$ is driven by SF interaction. Moreover, our {\it ab initio} calculations reveal that the SF interaction is an order of magnitude stronger than both EPC and Coulomb repulsion on the large quasi-two-dimensional FS pocket around the Brillouin zone (BZ) center, thus leading to the SF-mediated pairing mechanism, although the EPC dominates on the small three-dimensional electron FS pockets at the BZ corners. The emergence of nodal $d_{x^2-y^2}(\pm)$-wave gap structure is traced to the pronounced peaks in the Lindhard response function at the BZ corners. Our calculated FS, SC critical temperature, nodal gap structure and SC quasiparticle density of states are consistent with most available experiments. Furthermore, predicted unconventional SC properties such as scanning tunneling spectra of La$_{0.8}$Sr$_{0.2}$NiO$_2$ and Pr$_{0.8}$Sr$_{0.2}$NiO$_2$ are ready for immediate experimental verifications.