Doping dependence of magnetic excitations of one-dimensional cuprates as probed by resonant inelastic x-ray scattering

We study the dynamical, momentum-dependent two- and four-spin response functions in doped and undoped one-dimensional cuprates, as probed by resonant inelastic x-ray scattering, using an exact numerical diagonalization procedure. In the undoped t-J system, the four-spin response vanishes at ?, whereas the two-spin correlator is peaked around ?/2, with generally larger spectral weight. Upon doping, spectra tend to soften and broaden, with a transfer of spectral weight toward higher energy. However, the total spectral weight and average peak position of either response are only weakly affected by doping up to a concentration of 1/8. Only the two-spin response at ? changes strongly, with a large reduction of spectral weight and enhancement of excitation energy. At other momenta, the higher energy, generic features of the magnetic response are robust against doping. It signals the presence of strong short-range antiferromagnetic correlations, even after doping mobile holes into the system. We expect this to hold also in higher dimensions.