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We study the bilayer two-orbital model for superconducting pairing symmetryof La$_3$Ni$_2$O$_7$ under pressure. By combining density-functional-theory(DFT), maximally-localized-Wannier-function, and linearized Eliashberg equationwith random-phase-approximation, we find that the superconducting pairingsymmetry of La$_3$Ni$_2$O$_7$ is robustly $d_{xy}$ if its DFT band structure isaccurately reproduced in the downfolded model. We further show that fine-tuningof crystal-field-splitting between two Ni-$e_g$ orbitals qualitatively affectssuperconducting pairing symmetry of the bilayer two-orbital model, whichchanges from $d_{xy}$ to $s_{\pm}$ as the crystal-field-splitting exceeds acritical value. When the model only includes nearest-neighbor andsecond-nearest-neighbor hoppings, the crystal-field-splitting obtained byfitting to the DFT band structure is larger than the critical value and thusleads to $s_{\pm}$ superconducting pairing symmetry. When all nonzerolong-range-hoppings are also included in the model, the fittedcrystal-field-splitting is reduced and smaller than the critical value, whichmakes $d_{xy}$ superconducting pairing symmetry more favorable than $s_{\pm}$symmetry. Our work demonstrates that in downfolded effective models, thedetails of band structure can play a crucial role in determining pairingsymmetry in multi-orbital unconventional superconductors (such asLa$_3$Ni$_2$O$_7$).

Role of crystal-field-splitting and longe-range-hoppings on superconducting pairing symmetry of La$_3$Ni$_2$O$_7$