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We probe the collective magnetic modes of La$_2$CuO$_4$ (LCO) and underdopedLa$_{2-x}$Sr$_x$CuO$_4$ (LSCO) by momentum resolved Resonant Inelastic X-rayScattering (RIXS) at the Cu $L_3$ edge. In LCO the single magnon dispersionmeasured by RIXS coincides with the one determined by inelastic neutronscattering. For LSCO the spin dynamics shows a branch dispersing up to$\sim$400 meV coexisting with a branch at lower energy. Only the latter hasbeen observed with neutrons so far and is considered a key signature of dopedcuprates. The presence of the high-energy branch indicates that LSCO is in adynamic inhomogeneous spin state.

Evidence for Dynamic Phase Separation and High Energy Spin Dynamics in underdoped La$_{2-x}$Sr$_x$CuO$_4$ from Resonant Inelastic X-Ray Scattering