Theory of magnetic excitations in multilayer nickelate superconductor La$_{3}$Ni$_{2}$O$_{7}$

Motivated by the recent reports of high-$T_c$ superconductivity in La$_3$Ni$_2$O$_7$ under pressure, we analyzed theoretically the magnetic excitations in the normal and the superconducting state in this compound, which can be measured by inelastic neutron scattering or RIXS. We show that the bilayer structure of the spin response allows to elucidate the role of the interlayer interaction and the nature of the Cooper-pairing in a very efficient way. In particular, we demonstrate the key difference between the potential $s_\pm$ and $d$-wave gaps, proposed recently, by comparing the corresponding response in the even and odd channels of the spin susceptibility. We show that the mostly interlayer driven bonding-antibonding $s_\pm$ Cooper-pairing produces a single large spin resonance peak in the odd channel only near the $X$ point whereas spin resonances in both the odd and the even channel are predicted for the $d$-wave scenario.