Incommensurate spin fluctuations and competing pairing symmetries in La$_3$Ni$_2$O$_7$

The recent discovery of superconductivity in the bilayer Ruddlesden-Popper nickelate La$_3$Ni$_2$O$_7$ under high pressure has generated much interest in the superconducting pairing mechanism of nickelates. Despite extensive work, the superconducting pairing symmetry in La$_3$Ni$_2$O$_7$ remains unresolved, with conflicting results even for identical methods. We argue that different superconducting states in La$_3$Ni$_2$O$_7$ are in close competition and highly sensitive to the choice of interaction parameters as well as pressure-induced changes in the electronic structure. Our study uses a multi-orbital Hubbard model, incorporating all Ni $3d$ and O $2p$ states. We analyze the superconducting pairing mechanism of La$_3$Ni$_2$O$_7$ within the random phase approximation and find a transition between $d$-wave and sign-changing $s$-wave pairing states as a function of pressure and interaction parameters, which is driven by spin fluctuations with different wave vectors. These spin fluctuations with incommensurate wave vectors cooperatively stabilize a superconducting order parameter with $d_{x^2-y^2}$ symmetry for realistic model parameters. Simultaneously, their competition may be responsible for the absence of magnetic order in La$_3$Ni$_2$O$_7$, demonstrating that magnetic frustration and superconducting pairing can arise from the same set of incommensurate spin fluctuations.