Sensitive dependence of pairing symmetry on Ni-$e_g$ crystal field splitting in the nickelate superconductor La$_3$Ni$_2$O$_7$

The discovery of high-temperature superconductivity in La$_3$Ni$_2$O$_7$ under pressure has drawn great attention. However, consensus has not been reached on its pairing symmetry in theory. By combining density-functional-theory (DFT), maximally-localized-Wannier-function, and linearized gap equation with random-phase-approximation, we find that the pairing symmetry of La$_3$Ni$_2$O$_7$ is $d_{xy}$, if its DFT band structure is accurately reproduced by a downfolded bilayer two-orbital model. More importantly, we reveal that the pairing symmetry of La$_3$Ni$_2$O$_7$ sensitively depends on the crystal field splitting between two Ni-$e_g$ orbitals. A slight increase in Ni-$e_g$ crystal field splitting alters the pairing symmetry from $d_{xy}$ to $s_{\pm}$. Such a transition is associated with the change in inverse Fermi velocity and susceptibility, while the shape of Fermi surface remains almost unchanged. Our work highlights the sensitive dependence of pairing symmetry on low-energy electronic structure in multi-orbital superconductors, which calls for care in the downfolding procedure when one calculates their pairing symmetry.