Dispersion, damping, and intensity of spin excitations in the single-layer (Bi,Pb)$_{2}$(Sr,La)$_{2}$CuO$_{6+delta}$ cuprate superconductor family

Using Cu-$L_3$ edge resonant inelastic x-ray scattering (RIXS) we measured the dispersion and damping of spin excitations (magnons and paramagnons) in the high-$T_mathrm{c}$ superconductor (Bi,Pb)$_{2}$(Sr,La)$_{2}$CuO$_{6+delta}$ (Bi2201), for a large doping range across the phase diagram ($0.03lesssim plesssim0.21$). Selected measurements with full polarization analysis unambiguously demonstrate the spin-flip character of these excitations, even in the overdoped sample. We find that the undamped frequencies increase slightly with doping for all accessible momenta, while the damping grows rapidly, faster in the (0,0)$rightarrow$(0.5,0.5) nodal direction than in the (0,0)$rightarrow$(0.5,0) antinodal direction. We compare the experimental results to numerically exact determinant quantum Monte Carlo (DQMC) calculations that provide the spin dynamical structure factor $S(textbf{Q},omega)$ of the three-band Hubbard model. The theory reproduces well the momentum and doping dependence of the dispersions and spectral weights of magnetic excitations. These results provide compelling evidence that paramagnons, although increasingly damped, persist across the superconducting dome of the cuprate phase diagram; this implies that long range antiferromagnetic correlations are quickly washed away, while short range magnetic interactions are little affected by doping.