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Using Cu-$L_3$ edge resonant inelastic x-ray scattering (RIXS) we measuredthe dispersion and damping of spin excitations (magnons and paramagnons) in thehigh-$T_\mathrm{c}$ superconductor (Bi,Pb)$_{2}$(Sr,La)$_{2}$CuO$_{6+\delta}$(Bi2201), for a large doping range across the phase diagram ($0.03\lesssimp\lesssim0.21$). Selected measurements with full polarization analysisunambiguously demonstrate the spin-flip character of these excitations, even inthe overdoped sample. We find that the undamped frequencies increase slightlywith doping for all accessible momenta, while the damping grows rapidly, fasterin the (0,0)$\rightarrow$(0.5,0.5) nodal direction than in the(0,0)$\rightarrow$(0.5,0) antinodal direction. We compare the experimentalresults 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 reproduceswell the momentum and doping dependence of the dispersions and spectral weightsof magnetic excitations. These results provide compelling evidence thatparamagnons, although increasingly damped, persist across the superconductingdome of the cuprate phase diagram; this implies that long rangeantiferromagnetic correlations are quickly washed away, while short rangemagnetic interactions are little affected by doping.

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