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The collective magnetic excitations in the spin-orbit Mott insulator(Sr$_{1-x}$La$_x$)$_2$IrO$_4$ ($x=0,\,0.01,\,0.04,\, 0.1$) were investigated bymeans of resonant inelastic x-ray scattering. We report significant magnonenergy gaps at both the crystallographic and antiferromagnetic zone centers atall doping levels, along with a remarkably pronounced momentum-dependentlifetime broadening. The spin-wave gap is accounted for by a significantanisotropy in the interactions between $J_\text{eff}=1/2$ isospins, thusmarking the departure of Sr$_2$IrO$_4$ from the essentially isotropicHeisenberg model appropriate for the superconducting cuprates.

Anisotropic exchange and spin-wave damping in pure and electron-doped Sr$_2$IrO$_4$